Metrisable oscillators and (super)integrable two-dimensional metrics

Metrisable oscillators and (sup er)in tegrable t w o-dimensional metrics Jaume Gin ´ e 1 and Dmitry I. Sinelshc hiko v 2,3 1 Departamen t de Matem` atica, Univ ersitat de Lleida, Avda. Jaume I I, 69; 25001 Lleida, Catalonia, Spain 2 Instituto Biofisik a (UPV/EHU, CSIC), Univ ersit y of the Basque Coun try , Leioa E-48940, Spain 3 Ik erbasque F oundation, Bilbao 48013, Spain Marc h 31, 2026 Abstract W e consider a family of nonlinear oscillators, whic h is the autonomous case of the tw o- dimensional pro jective connection. W e construct sev eral classes of these oscillators that are sim ultaneously in tegrable and metrisable. This leads to families of (sup er)in tegrable t wo- dimensional metrics that are parametrized b y arbitrary functions. In the sup erin tegrable case we obtain an explicit expression for the unparametrized geo desics. In the integrable case w e present t wo families of metrics with transcendental first integrals. W e introduce the concept of generalized Darb oux in tegrability in the con text of b oth pro jectiv e equations and geodesic flows. W e demonstrate that the constructed in tegrable metrics are generalized Darb oux in tegrable. In addition, w e establish a direct connection b et ween relativ e Killing v ectors and inv ariants of the pro jective vector fields that are linear in the first deriv ativ e. Finally , w e compute the dimensions of the pro jectiv e Lie algebra for the obtained metrics, whic h allo ws us to distinguish previously known in tegrable cases from new ones. 1 In tro duction The (sup er)in tegrabilit y of tw o-dimensional geo desic flo ws on Riemannian manifolds has b een extensiv ely studied b oth in mathematical ph ysics and differential geometry [1–4]. How ev er there are still op en problems in this area. F or example, there are op en problems that are connected with the existence of polynomial and rational first integrals [2–6] or explicit expressions for the geo desics [3, 4]. F urthermore, there has b een in terest in constructing (sup er)in tegrable metrics with polynomial and rational first in tegrals (see, e.g. [5–13]). F or example, in [5] it w as demonstrated that there exist metrics with rational in tegrals of any degree. In [12–14] it is prov ed that for an y metric the space of rational integrals is finite-dimensional and the connection b et ween relative Killing tensor fields and rational in tegrals is discussed. Recen tly , the connections b etw een in tegrability of nonlinear oscillators and geo desic flows and vice v ersa [15–17] hav e b een studied. In this work, we extend the results obtained in [15] and 1 study the metrisability of nonlinear oscillators that can b e linearized with the help of nonlo cal transformations [18, 19]. W e prop ose a new approac h for constructing (sup er)in tegrable metrics from nonlinear oscillators, whic h is based on finding in tersections b et w een integrabilit y and metris- abilit y conditions. The main idea b ehind this approac h is that nonlo cal transformations neither preserv e metrisabilit y nor Lie symmetries of the pro jectiv e connection, but preserve b oth its form and integrabilit y . This allows one to deform a trivially metrisable pro jective equation, for example a linear equation, and obtain new non trivially metrisable and in tegrable pro jective connections. The linearizability conditions for the t wo-dimensional autonomous pro jective connections via generalized nonlo cal transformations split in to three cases, eac h of which pro vides a family of in tegrable metrics parametrized by arbitrary functions. F or the first linearizabilit y condition we demonstrate that there is a family of sup erin tegrable metrics with a linear and a transcenden tal in momenta first in tegrals. W e pro vide a completed classification the pro jective Lie algebras of this family of metrics. As a byproduct of this result, w e sho w that for metrisable equations obtained in [16] this Lie algebra is either sl (2 , R ) or sl (3 , R ). This means that these in tegrable metrics are either Darboux-sup erintegrable (i.e. they ha ve four linearly indep enden t second-degree p olynomial first in tegrals, see e.g. [1]) or flat, i.e. hav e a constan t curv ature. This app ears to b e the first time when the complete classification of the pro jectiv e Lie algebras for linearizable via nonlo cal transformations oscillators is carried out. F or the second and third linearizability conditions w e obtain tw o new integrable families of the conformal metrics with, in general, transcendental first in tegrals. W e demonstrate that these first in tegrals are Darb oux functions (see, e.g. [20, 21] for the definition) of in v arian ts of the pro jectiv e connection that are p olynomial in the first deriv ative and of relativ e Killing v ectors for the geodesic flo w. This allo ws us to introduce the concept of the generalized Darb oux integrabilit y for b oth pro jective v ector fields and Hamiltonian geo desic flo ws. In con trast to the classical Darboux in tegrability theory (see, e.g. [20–22]), where Darb oux first in tegrals are formed by the p olynomial in v ariants, w e allo w the in v ariants to b e polynomials only in a subset of phase v ariables (see also [23–25]). W e also demonstrate that there is a direct connection b etw een the relativ e Killing v ectors of geodesic flo ws and the in v arian ts of pro jective vector fields that are linear in the first deriv ativ e. Finally , we obtain that for particular v alues of the parameters, one of the ab ov e transcendental first integrals can b e reduced to a rational in tegral of an arbitrary degree. It is also w orth noting that the conditions that define these in tegrable conformal metrics are themselv es in tegrable second- order differen tial equations. F urthermore, we pro ve that in the generic case of these metrics their pro jective Lie algebra is one-dimensional. The rest of this w ork is organized as follows. In the next Section we provide basic definitions that concern in tegrabilit y of t wo-dimensional Riemannian metrics and nonlinear oscillators. W e also pro vide direct form ulas that connect the relativ e Killing vectors and linear in the first deriv ativ e in v ariants of pro jective v ector fields. In Section 3 w e discuss the family of linearizable via nonlo cal transformations cubic oscillators, construct their autonomous and non-autonomous first in tegrals and classify their Lie symmetries. W e devote Section 4 to the construction of sup erin tegrable metrics with a linear and a transcendental first integrals. In Section 5 w e construct in tegrable families of the conformal metrics and introduce the notion of generalized Darb oux integrabilit y . In the last Section w e briefly summarize and discuss our results. 2 2 Pro jectiv e equation and its in tegrabilit y W e b egin by briefly in tro ducing the definitions and concepts that are used in the subsequent sections. Consider a smooth t w o-dimensional manifold M with the (pseudo)Riemannian metric g = ( g ij ), i, j = 1 , 2, g ij = g j i . The parametrized geo desics of g on T M are solutions of ¨ x i + Γ i j k ˙ x j ˙ x k = 0 , ( x 1 , x 2 ) = ( x, y ) , (2.1) where Γ i j k is the Levi-Civita connection [26] for the metric g . Notice that here and b elow the summation con v ention o ver rep eated indices is assumed. W e also denote by X Γ = v i ∂ x i − Γ i j k v j v k ∂ v i , v i = ˙ x i the geo desic spra y . On T ∗ M system (2.1) can b e presented in the Hamiltonian form: ˙ x i = H p i , ˙ p i = − H x i , H = 1 2 g ij p i p j , g ik g kj = δ i j . (2.2) One can pro ject (2.1) on the ( x, y ) plane to obtain y xx + a 3 ( x, y ) y 3 x + a 2 ( x, y ) y 2 x + a 1 ( x, y ) y x + a 0 ( x, y ) = 0 , (2.3) where a 3 = − Γ 1 22 , a 2 = Γ 2 22 − 2Γ 1 12 , a 1 = 2Γ 2 12 − Γ 1 11 , a 0 = Γ 2 11 . (2.4) Therefore, for every metric there exists an equation from (2.3) that is a pro jection of its geo desic flo w. Ho wev er, the con verse of this statement is not true, i.e. not every equation from (2.3) corre- sp onds to some metric. This results from the fact that there are four co efficien ts of the pro jectiv e connection (2.3) and only three comp onents of the metric g . Equations from (2.3) that corre- sp ond to a metric are called metrisable and necessary and sufficient conditions for metrisability of (2.3) w ere obtained in [27]. Notice that if (2.3) is metrisable then its solutions are unparam- eterized geo desics of the corresponding metric. The vector filed asso ciated to (2.3) is denoted X p = ∂ x + u∂ y − P 3 j =0 ( a j u j ) ∂ u , u = y x . In this work we study metrisabilit y of a family of cubic oscillators, whic h is a particular case of (2.3) and is giv en by y xx + k ( y ) y 3 x + h ( y ) y 2 x + f ( y ) y x + g ( y ) = 0 . (2.5) Here k ( y ), h ( y ), f ( y ) and g ( y ) are some sufficien tly smo oth functions. Throughout this w ork we assume that g ( y )  = 0. The following vector filed X = u∂ y − ( k u 3 + hu 2 + f u + g ) ∂ u , whic h is x indep enden t case of X p , is asso ciated to (2.5). W e also remark that the restriction to the autonomous case of the pro jectiv e connection does not imply that the corresp onding metric is indep enden t of x . While the metrisability conditions for the general case of (2.5) are cum b ersome and follo w from the results of [27], it is w orth considering particular cases of (2.5) that admit an explicit (p ossibly nonautonomous) first integral. Suc h in tegral can b e lifted to a first integral of (2.2). If it is functionally indep endent of the Hamiltonian, this establishes the (sup er)integrabilit y of (2.2). Moreo ver, this idea can also b e used in the reverse order, when first in tegrals of (2.2) are pro jected to demonstrate the integrabilit y of an equation from (2.5). Consequently , finding the intersections of integrabilit y and metrisabilit y conditions for (2.5) in particular, or more generally for (2.3), pro vides a mechanism for generating (sup er)integrable metrics. Conv ersely , one can pro ject the in tegrals of (2.2) onto the ( x, y ) plane to prov e the integrabilit y of an equation from (2.3) or (2.5). 3 Belo w, to study the metrisabilit y of in tegrable cases of (2.5) we transform (2.4) in to the Liouville system [1, 27, 28]. Using the Christoffel sym b ols (see, e.g. [26]) to connect (2.4) to the metric comp onen ts, and then applying the transformations ψ 1 = ∆ 2 g 11 , ψ 2 = ∆ 2 g 12 , ψ 3 = ∆ 2 g 22 , ∆ = ψ 1 ψ 3 − ψ 2 2  = 0 , (2.6) w e obtain ψ 1 ,x = − 2 3 a 1 ψ 1 + 2 a 0 ψ 2 , ψ 3 ,y = − 2 a 3 ψ 2 + 2 3 a 2 ψ 3 , ψ 1 ,y + 2 ψ 2 ,x = − 4 3 a 2 ψ 1 + 2 3 a 1 ψ 2 + 2 a 0 ψ 3 , ψ 3 ,x + 2 ψ 2 ,y = − 2 a 3 ψ 1 + 4 3 a 1 ψ 3 − 2 3 a 2 ψ 2 . (2.7) This is a linear ov erdetermined system of four equations for three functions ψ m , m = 1 , 2 , 3. An equation from (2.3) is metrisable if and only if there is a solution of (2.7) suc h that ∆  = 0 [1, 27–29]. F ollo wing [1, 30] w e denote the Lie algebra of p oint symmetries of (2.5) as p ( c ) in the general case and p ( g ) when the corresp onding pro jective structure is metrisable. This Lie algebra is called the pro jective Lie algebra. It is well kno wn [31–34] that for a general pro jectiv e structure (2.3) the dimension of p ( c ) belongs to { 0 , 1 , 2 , 3 , 8 } . F urther w e compute the dimensions of p ( g ) for metrisable cases of (2.5). Recall also that if dim p ( g ) = 3 then the metric is Darb oux sup erintegrable and if dim p ( g ) = 8 then the metric is flat and the corresp onding pro jectiv e connection can b e linearized via the p oin t transformations [1, 30]. Let us in tro duce the basic definitions that concern the in tegrability of the geo desic flo w and its pro jection. A smo oth function J = J ( x, y , v 1 , v 2 ) that satisfies the relation X Γ J = N J for some N = N ( x, y , v 1 , v 2 ), whic h is called the cofactor, is an in v arian t of the geo desic spra y . W e can introduce the inv ariants of the general pro jectiv e connection or its autonomous case in the same wa y as solutions of X p R = l R , R = R ( x, y , u ) for some cofactor l = l ( x, y , u ) or X R = l R , R = R ( y , u ) for some cofactor l = l ( y , u ). A first in tegral is an inv ariant with zero cofactor. F or (2.5) w e also use the notion of an non-autonomous first in tegral whic h is a smo oth function satisfying R x + X R = 0 and an integrating factor M ( y , u ), which is a smooth function that is a solution of the equation div( M X ) = 0. If there is a global, possibly except on a set of zero Leb esgue measure, explicit expression for the first integral or an integrating factor of autonomous pro jective equation (2.5), the corresp onding t wo-dimensional dynamical system is called completely in tegrable [20–22]. Let {· , ·} b e the canonical Poisson brac k et. Then, a smo oth function T ( x, y , p 1 , p 2 ) is a first in tegral of (2.2) if { T , H } = X H T = 0. Hamiltonian system (2.2) is integrable in the Liouville- Arnold sense if, apart from the Hamiltonian, there is an additional first integral, functionally indep enden t of H , whic h is in inv olution with H . Hamiltonian system (2.2) is sup erin tegrable if there are t w o first integrals of (2.2) that are in in volution with H and pairwise functionally indep enden t. First integrals of the pro jective equation (2.5) are related to the first integrals of the geo desic flow (2.2) via the lift y x = u = H p 2 /H p 1 . The inv erse statement is true for the first in tegrals of (2.2) that are homogeneous functions of the momenta. Recen tly , in [12, 13] the concept of the relative Killing tensors w as introduced to study rational first integrals of geo desic flows (see also [35, 36]). A p olynomial in momen ta function of degree d is called a relativ e Killing tensor if { H, K } = LK , where L is a necessary linear in momen ta function. Geometrically this means that the Hamiltonian vecto r field is tangen t to the submanifold K = 0 on T ∗ M . 4 Belo w w e demonstrate that the first integrals obtained in Section 5 are transcenden tal f unctions of the homogeneous relative Killing tensors of degree 1 or homogeneous relative Killing vectors. Suc h in tegrals generalize functional class of rational first in tegrals to the class of the generalized Darb oux functions. Finally , we establish a direct corresp ondence b etw een homogeneous relative Killing v ectors and p olynomial with resp ect to y x = u in v arian ts of the pro jective vector fields. Prop osition 2.1. Supp ose that the c o efficients of (2.3) satisfy (2.4) for some metric g and also e quation (2.3) has an invariant Z = P 2 i =1 e i ( x, y ) u i − 1 , e i ∈ C ∞ ( M ) with the c ofactor l = P 3 m =1 b m ( x, y ) u m − 1 , b m ∈ C ∞ ( M ) . Then the ge o desic spr ay X Γ and Hamiltonian (2.2) b oth have a homo gene ous, line ar in velo cities, invariant and a homo gene ous r elative Kil ling ve ctor, r esp e ctively, which have the form J = e i v i , N = b i v i − Γ 1 1 i v i − Γ 1 12 v 2 , (2.8) K = e k p k , L = Γ 1 1 j g j k p k + Γ 1 12 g 2 k p k − b k p k , e k = g kj e j , b k = g kj b j . (2.9) Inversely, if Hamiltonian system (2.2) has a homo gene ous r elative Kil ling ve ctor, then e quation (2.3) whose c o efficients satisfy (2.4) has an invariant that is line ar in y x = u with a quadr atic in y x = u c ofactor. Pr o of. Suppose that co efficients of (2.3) satisfy (2.4) for some metric g and also equation (2.3) has an inv ariant Z = P 2 i =1 e i ( x, y ) u i − 1 with the cofactor l = P 3 m =1 b m ( x, y ) u m − 1 . Substituting the expression for the in v ariant and the cofactor in the resp ective definition and collecting co efficien ts at the same p o wers of u one can find that b 3 = Γ 1 22 . Consider ˜ J = Z | z = v u and ˜ N = v 1 l | z = v 2 v 1 . Then, compute X Γ ˜ J = ˜ N ˜ J and X R = lR collecting co efficien ts at the p o wers of v j and u , resp ectively . Comparing the equations for e i and b m obtained in this wa y one can demonstrate that if X R = l R is satisfied, then so is X Γ ˜ J = ˜ N ˜ J and vice v ersa. Both ˜ J and ˜ N are rational functions of v 1 . Note also that v 1 is the inv arian t of X Γ with the cofactor − (Γ 1 j k v j v k ) /v 1 . Then consider the in v arian t J = ˜ J v 1 = e i v i , whic h is a linear homogeneous p olynomial in v i , i = 1 , 2. The corresp onding cofactor has the form N = b i v i − Γ 1 1 i v i − Γ 1 12 v 2 (recall that b 3 = Γ 1 22 ). As a consequence, we ha ve that the geodesic spray X Γ has a linear homogeneous in v ariant in velocities with a linear homogeneous in v elo cities cofactor. Finally , it is kno wn that the Legendre transformations v j = g j k p k map the geo desic spray in to the Hamiltonian flo w {· , H } (see, e.g. [37]). Recall also that in the definition of the relative Killing v ector the flo w { H, ·} = −{· , H } is used and, consequen tly , the cofactor changes the sign. As a result, w e obtain expressions (2.9). The consideration abov e can b e repeated in the rev erse order to demonstrate that for an y Killing v ector of (2.2) there is an inv ariant that is linear in u = y x of the metrisable pro jective structure. This completes the pro of. Prop osition 2.1 is used in Section 5 to map inv ariants of (2.5), which are linear in u = y x , to the relativ e Killing vectors of integrable Hamiltonian geo desic flows (2.2). 3 Cubic oscillators linearizable via nonlo cal transforma- tions and pro jectiv e symmetries In this section w e briefly presen t the results on the linearizabilit y of family (2.5) with resp ect to nonlo cal transformations obtained in [18, 19]. W e demonstrate that the linearizability is equiv alent 5 to the existence of a cubic rational, with resp ect to u , integrating factor and a first integral, and, in a particular case, a non-autonomous first integrals. As a linear equation w e consider w ξ ξ + β w ξ + α w = 0 . (3.1) Here αβ  = 0 are arbitrary parameters. W e use the following family of nonlo cal transformations w = F ( y ) , dξ = [ G 1 ( y ) y x + G 2 ( y )] dx, (3.2) where it is assumed that F , G 1 and G 2 are sufficien tly smo oth functions and F y G 1 G 2  = 0, since otherwise transformations (3.2) either degenerate or are reduced to p oin t or generalized Sundman ones. T ransformations (3.2) are a com bination of p oin t, con tact, and Sundman transformations. They preserve autonomous pro jective structures, i.e., the family of equations (2.5) is closed under (3.2). If G 2 = 0 then (3.2) reduce to p oint transformations. If G 1 = 0, then (3.2) are generalized Sundman transformations and if, in addition, F ( y ) = y then (3.2) are Sundman transformations. It is clear that point transfo rmations preserv e b oth the metrisability and the pro jectiv e Lie algebra of (2.5). Ho wev er, the Sundman transformations, as a special case, and transformations (3.2) in general, neither preserv e the metrisability of (2.5) nor its pro jectiv e Lie algebra (see, e.g. [38]). This allo ws us to prop ose the following approach for constructing integrable pro jective struc- tures. Supp ose w e start from a trivially metrisable and integrable pro jective v ector field, for in- stance, a linear equation, and construct its equiv alence class with resp ect to transformations (3.2). T ransformations (3.2) preserve the in tegrabilit y of (2.5), since they map autonomous first in tegrals to autonomous first in tegrals [39]. Ho wev er, metrisabilit y is not preserv ed, and thus w e need to find the in tersection b etw een the equiv alence class and metrisabilit y conditions. If this intersection is not empty , w e obtain families of pro jective structures that are b oth in tegrable and metrisable. Finally , we lift the corresp onding first in tegral and/or in v arian t of the pro jective connection to a first in tegral and/or inv ariant of the geo desic flo w. Here. we apply this approac h to linear equa- tion (3.1) and to the autonomous case of the pro jective connection (2.5). W e plan to consider the general case of the pro jectiv e connections that are linearizable via nonlo cal transformations elsewhere. F urther we need the following simple result: Prop osition 3.1. Line ar e quation (3.1) p ossesses two first or der p olynomial invariant curves e Z 1 , 2 = 2 v + ( β ± p β 2 − 4 α ) w , λ 1 , 2 = − β ∓ p β 2 − 4 α 2 . (3.3) If, in addition the fol lowing r elation on the p ar ameters α , β holds 2 β 2 − 9 α = 0 , (3.4) then (3.1) has a se c ond or der p olynomial invariant curve e Z 3 = c 1 e Z 2 1 + c 2 e Z 2 , λ 3 = − 2 β 3 , c 1 c 2  = 0 . (3.5) The inte gr ating factor of (3.1) c orr esp onding to (3.3) has the form f M = ( e Z 1 e Z 2 ) − 1 , (3.6) 6 while the first inte gr al is e R = e Z √ β 2 − 4 α + β 1 e Z √ β 2 − 4 α − β 2 . (3.7) We use the notation v = w ξ in al l of the ab ove expr essions. Pr o of. It is not difficult to demonstrate that the highest degree of an in v arian t curve of (3.1) is 2. It is also clear that the cofactor is a constant, since the degree of the v ector field asso ciated to (3.1) is 1. Thus, it is straightforw ard to v erify that the only p ossible first order inv ariant curves of (3.1) are (3.3) and an irreducible second order inv ariant exists if and only if (3.4) holds and has the from (3.5). In tegrating factor (3.6) and first integral (3.7) can b e directly found from inv arian t curv es (3.3) This completes the pro of. No w w e presen t equiv alence criterion for (2.3) and linear equation (3.1) via transformations (3.2). Below we use the following notations to simplify the presentation of the results A = 27 k g 2 − 9 hf g + 2 f 3 + 9 g f y − 9 f g y , B = 3 g h − f 2 + 3 g y . (3.8) The follo wing prop osition holds (see Theorem 2.1 and Corollary 2.1 in [18]): Prop osition 3.2. Supp ose that g  = 0 . Then the fol lowing statements ar e e quivalent: 1) e quation (2.5) c an b e tr ansforme d into (3.1) via (3.2) , wher e the functions F and G ar e given by F y = αF G 2 2 g , G 1 = G 2 ( f − β G 2 ) 3 g , (3.9) and G 2 satisfies by one of the fol lowing c orr elations 1a) G 2 = ( β 2 − 3 α ) A 2 β (2 β 2 − 9 α )( g A y − B A ) , A ( β 2 − 3 α )(2 β 2 − 9 α )( g A y − B A )  = 0; (3.10) 1b) G 3 2 = − A β 3 , β 2 − 3 α = 0 , A  = 0; (3.11) 1c) G 2 ,y = G 2 ( β 2 G 2 2 + 3 B ) 9 g , 2 β 2 − 9 α = 0 , (3.12) 2) the functions k , h , f , g and the p ar ameters α and β satisfy one of the r elations 2a) β 2 (2 β 2 − 9 α ) 2 ( g A y − B A ) 3 − A 5 ( β 2 − 3 α ) 3 = 0 , (2 β 2 − 9 α )( β 2 − 3 α ) A ( g A y − B A )  = 0; (3.13) 2b) β 2 − 3 α = 0 , (3.14) g A y − B A = 0; (3.15) 7 2c) 2 β 2 − 9 α = 0 , (3.16) A = 0; (3.17) 3) e quation (2.5) has a r ational and cubic with r esp e ct to u = y x inte gr ating factor M = g G 2 2 [( f − β G 2 ) u + 3 g ][(2 β − 9 α 2 ) G 2 2 u 2 − β G 2 ( f u + 3 g ) u − ( f u + 3 g ) 2 ] ; (3.18) 4) e quation (2.5) p ossesses a first inte gr al R = F 2 ρ [( f − β G 2 ) u + 3 g ] − 2 ρ (6 αG 2 u + ( β + ρ )[( f − β G 2 ) u + 3 g ]) ρ + β × (6 αG 2 u + ( β − ρ )[( f − β G 2 ) u + 3 g ]) ρ − β , ρ = p β 2 − 4 α (3.19) It is also worth noting that correlation (3.12) is the Bernoulli differen tial equation for the function G 2 , whic h can be explicitly solv ed. W e do not provide the corresponding expression in order to simplify the presen tation of the results. In what follo ws, we also use the notation B = 3 2  g y + g C y y C y  , C y  = 0 , (3.20) whic h, in terms of the functions h , f and g is equiv alent to C = Z g ( y ) exp  2 Z  h − f 2 3 g  dy  dy . (3.21) The follo wing statement holds: Corollary 3.1. If c ondition (3.4) holds, then line arizable via (3.2) e quations have a r ational cubic inte gr ating factor M = g G 2 2 ( f u + 3 g )([ β G 2 + f ] u + 3 g )([ β G 2 − f ] u − 3 g ) , (3.22) r ational quadr atic with r esp e ct to u first inte gr al R = 9 g C y u 2 ( f u + 3 g ) 2 + 2 C , (3.23) and the non-autonomous first inte gr al R = y Z 0 f ( ω ) 3 g ( ω ) − s C ′ ( ξ ) g ( ω )[ R − 2 C ( ω )] ! dω + x, (3.24) wher e R is given in (3.23) and ω is an inte gr ation variable. No w w e obtain a classification of possible dimensions of Lie algebras admitted b y (2.5) when condition (3.17) is satisfied. This classification is used b elow to find cases of the Darb oux sup erin- tegrable and flat metrics that corresp ond to (2.5) under (3.17). T o this aim, w e use the results of classification of Lie p oint symmetries of the pro jectiv e equations given in [34]. In [34] all equations 8 of the from (2.3) are classified in terms of their relativ e and basis differential in v ariants of p oint transformations. The dimension of the admitted Lie algebra can b e determined b y basis algebraic in v ariants. Below w e compute relativ e and absolute inv ariants for (2.3) when condition (3.17) is met. Recall also that pro jective equation (2.3) can b e linearized via p oint transformations and dim p ( c ) = 8 if and only if the Liouville in v arian ts L 1 and L 2 v anish sim ultaneously [28, 31–33]. F or equation (2.5) the Liouville in v arian ts ha ve a v ery compact represen tation in terms of the functions A and B defined in (3.8): L 1 = − B y 3 , L 2 = A y − 3 f B y 27 g , (3.25) whic h is used in the pro of of the following result: Theorem 3.1. Supp ose that c orr elation (3.17) holds and g  = 0 . Then dim p ( c ) c an b e one, two, thr e e or eight. In the generic c ase dim p ( c ) = 1 . We have that dim p ( c ) = 2 if and only if g B B y y + g y B B y − 2 g B 2 y = 0 , g B y (9 g B y − 4 B 2 )  = 0 . (3.26) We have that dim p ( c ) = 3 and p ( c ) is isomorphic to sl (2 , R ) if and only if 9 g 2 B 2 y y + 9 g ( B + 2 g y ) B y B y y − 25 g B 3 y − (4 B + 3 g y ) ( B − 3 g y ) B 2 y = 0 , g B y  = 0 , (3.27) 2 g B y y + B B y + 2 g y B y  = 0 . (3.28) Final ly, dim p ( c ) = 8 and p ( c ) is isomorphic to sl (3 , R ) if and only if B y = 0 , g  = 0 . (3.29) Pr o of. In order to obtain the pro of of this statement w e use Theorems 1 and 4 from [34]. Notice that in [34] instead of the Liouville inv ariants the notations β 1 = −L 1 and β 2 = −L 2 are used. W e will stic k to the classical notation and use expressions (3.25) for the Liouville in v arian ts. In what follo ws w e also express in v ariants and other quantities for (2.3) in terms of the functions A and B (see definitions in (3.8)). All other notation is inherited from [34]. Since (2.3) do es not dep end explicitly on x , it is clear that dim p ( c ) ∈ { 1 , 2 , 3 , 8 } . Belo w, we demonstrate that all four cases are p ossible. W e use the follo wing criteria for differen t dimensions of p ( c ). According to [34] (see Theorems 1 and 4 there) w e ha v e that: 1) dim p ( c ) = 2 if and only if all algebraic basis in v arian ts of (2.3) are constant; 2) dim p ( c ) = 3 if and only if J 0 = j 0 = j 1 = j 2 = Γ 0 j 3 + 5 L 1 = 0 and L 1  = 0. It is w ell known that dim p ( c ) = 8 if and only if L 1 = L 2 = 0. Supp ose that (3.17) holds. Then using Theorem 1 of [34] w e compute the quantities J 0 , j k , k = 0 , 1 , 2 , 3, Γ 0 , Λ and e 0 for (2.3). W e also substitute condition (3.17) into the definition of the Liouville in v ariants (3.25) to obtain expressions for L 1 and L 2 . As a result, we find that the relativ e in v arian t J 0 and the functions j 0 , j 2 and e 0 v anish. The rest of the ab ov e quantities are 9 giv en by j 1 = − 1 486 g 2  45 g 2 B y B y y y − 54 g 2 B 2 y y − 9 g B y ( B − 3 g y ) B y y − 15 B 3 y g +  4 B 2 − 9 B g y + 45 g g y y − 9 g 2 y  B 2 y  , j 3 = − 9 (4 B B y + 3 g B y y + 3 B y g y ) 5 B 3 y , Γ 0 = B y (3 g B y y + 3 B y g y − B B y ) 27 g , (3.30) Λ = 9 25 B 6 y  135 B y B y y y g 2 + 252 B 2 y y g 2 + 3 B y g (374 B + 303 g y ) B y ,y + B 2 y  808 B 2 + 1122 B g y + 30 g B y + 135 g g y ,y + 387 g 2 y   , L 1 = B y 3 , L 2 = f B y 9 g . W e see that L 1 j 1  = 0 for arbitrary functions f , g , h and k satisfying (3.17). Therefore, a generic equation from (2.3) satisfying (3.17) falls into the fourth class of equations listed in [34]. Then, taking in to account that j 2 = 0, w e compute basis algebraic in v arian ts I 1 = − Γ 0 L 1 j 1 / 2 1 , I 2 = 10 j 1 / 2 1 j 3 , L 1 j 1  = 0 . (3.31) As a result, w e find that I 1 and I 2 are not constant for arbitrary c hoice of functions f , g , h and k satisfying (3.17). Therefore, in the generic case, the Lie algebra of p oint symmetries of (2.3) satisfying (3.17) is one dimensional. Let us also remark that when j 1 = 0 and j 3 L 1 (Γ 0 j 3 − 5 β 1 )  = 0, an equation from (2.3) whose co efficien ts fulfill (3.17) falls in to the sixth class of equations listed in [34]. The corresp onding basis relative inv ariants are I 1 = − Γ 0 j 3 L 1 , I 2 = (5 − I 1 )Λ 5 j 2 3 , L 1 j 3 (Γ 0 j 3 + 5 L 1 )  = 0 , (3.32) where we use the fact that e 0 = 0. The in v arian ts given in (3.32) are also non-constant for an arbitrary c hoice of unctions f , g , h and k satisfying (3.17). Consequen tly , we see again that generically dim p ( c ) = 1. Let us classify equations from (2.3) that satisfy (3.17) and ha v e that dim p ( c ) = 2. First, w e supp ose that L 1 j 1  = 0 and require that I 1 ,y = I 2 ,y = 0 for I 1 , I 2 pro vided in (3.31). As a result, w e obtain the following system of equations L 1 j 1 Γ 0 ,y − Γ 0 j 1 L 1 ,y − 1 2 Γ 0 L 1 j 1 ,y = 0 , j 3 j 1 ,y + 2 j 1 j 3 ,y = 0 . (3.33) Assume that j 3  = 0. Then, excluding j 1 from the first equation of (3.33) via the second one w e get j 3 L 1 Γ 0 ,y − Γ 0 j 3 L 1 ,y + Γ 0 L 1 j 3 ,y = 0 . (3.34) If w e substitute the expressions for j 3 , L 1 and Γ 0 in to (3.34), we obtain a third-order differen tial equation with resp ect to B . The co efficien t in front of B y y y has the form C 3 = 2 g B y y + B B y + 2 g y B y . (3.35) 10 W e supp ose that C 3  = 0 and substitute the expression for B y y y obtained from (3.34) in to the second equation of (3.33). As a consequence, we get 625 B 5 y g 2 Γ 2 0 j 3 ( j 2 3 B 3 y + 9 j 3 B B y − 81 g )( g B B y y + g y B B y − 2 g B 2 y ) 9 C 3 3 = 0 . (3.36) First, w e supp ose that g B B y y + g y B B y − 2 g B 2 y = 0, whic h mean t that the condition from (3.26) holds. One can verify that under this conditions b oth equations in (3.33) v anish. Moreo ver, in tegrating (3.26) once w e get g B y − c 1 B 2 = 0 , (3.37) where c 1 is an in tegration constant. Using b oth (3.26) and (3.37) one can demonstrate that I 1 and I 2 are constants. Ho w ev er, the function j 1 v anishes when c 1 = 1 / 6 or c 1 = 4 / 9, whic h is the same as (9 g B y − 4 B 2 )(6 g B y − B 2 ) = 0. These cases will b e taken into account b elo w when we consider the case of j 1 = 0. Moreov er, from the condition Γ 0 = 0 it follo ws that j 1 = 0 and, hence, with this case we also deal b elo w. W e are only left with the p ossibility of j 2 3 B 3 y + 9 j 3 B B y − 81 g = 0, whic h leads to j 1 = 0 as well. If w e assume that C 3 = 0 w e obtain that b oth equations in (3.33) v anish if and only if B 2 + 4 g B y = 0, whic h is a particular case of (3.37). Therefore, the case of C 3 = 0 is included in to (3.26). Finally , one can show that j 3 and the first equation from (3.33) v anish sim ultaneously if and only if 2 B 2 + 3 g B y = 0, whic h is again a particular case of (3.26). As a consequence, w e obtain that all equations from (2.5), whose coefficients satisfy (3.17), ha ve dim p ( c ) = 2 in the case of j 1  = 0 if and only if (3.26) is fulfilled provided that (9 g B y − 4 B 2 )(6 g B y − B 2 )  = 0. Below we sho w that the condition 6 g B y − B 2  = 0 can b e remov ed since it corresp onds to the equations from (2.5) under condition (3.17), whic h falls into the sixth class of equations listed in [34] and having t wo- dimensional Lie algebra of p oin t symmetries. If 9 gB y − 4 B 2 = 0 w e obtain that j 1 = Γ 0 j 3 + 5 L 1 = 0 and the corresp onding equations from (2.3) hav e dim p ( c ) = 3 and this case is considered b elow separately . In order to finish with the equations from (2.3) having t wo-dimensional Lie algebra w e consider the case of j 1 = 0 assuming that L 1 j 3 (Γ 0 j 3 + 5 L 1 )  = 0. Otherwise, we will ha ve that dim p ( c ) is 3 or 8. In the same w ay as in the case of j 1  = 0 w e differentiate basis in v arian ts giv en in (3.32) and require that these deriv ative are zeros. The corresp onding computations are similar to those of the case of j 1  = 0 and, th us, are omitted. The only condition that fulfills that L 1 j 3 (Γ 0 j 3 + 5 L 1 )  = 0 and I 1 ,y = I 2 ,y = 0 is 6 g B y − B 2 = 0. This is a particular case of (3.37) and, hence, of (3.26). Therefore, as a criterion for (2.3) under condition (3.17) to hav e dim p ( c ) = 2 w e can use (3.26), taking into accoun t that if 6 g B y − B 2  = 0 or 6 g B y − B 2 = 0 the basis inv ariants are giv en b y (3.31) or (3.32), resp ectiv ely . No w we consider the case of a three-dimensional Lie algebra of p oint symmetries. T aking into accoun t that j 0 = j 2 = 0 for (2.3) under (3.17), we hav e that according to Theorem 1 of [34] dim p ( c ) = 3 if and only if j 1 = Γ 0 j 3 + 5 L 1 = 0, pro vided that L 1  = 0. Condition Γ 0 j 3 − 5 β 1 = 0 is equiv alen t (3.27). Moreov er, if we differentiate (3.27) once, we get 9 g 2 B y C 3 B y y y − 18 g 3 B 3 y ,y − g  16 g B y − 9 B 2 − 27 B g y − 18 g g y y − 18 g 2 y  B 2 y B y y −  4 B 3 − 9 B 2 g y + 33 B g B y − 9 B g g y y − 9 B g 2 y + 16 g B y g y − 18 g g y g y y  B 3 y = 0 , (3.38) where C 3 is giv en b y (3.35). If C 3 = 0, then Γ 0 j 3 + 5 L 1 and j 1 v anish simultaneously if and only if B = 0, whic h is a particular case of condition (3.29) for dim p ( c ) = 8. Th us, we assume that C 3  = 0 and exclude B y y y from j 1 via (3.38) and then use (3.27) to exclude B 2 y y . As a result, w e obtain that j 1 = 0. Therefore, Γ 0 j 3 + 5 L 1 = 0 yields the condition j 1 = 0 if C 3  = 0. As a consequence, 11 w e ha ve that equations from family (2.3) that satisfy (3.17) ha v e a three-dimensional pro jectiv e Lie algebra if and only if condition (3.27) is fulfilled. Moreo v er, it is known (see, e.g. [32, 34]) that this three dimensional Lie algebra is sl (2 , R ). Finally , an equation from (2.3) can ha ve an eigh t-dimensional pro jective Lie algebra if and only if both Liouville in v arian ts v anish. Using the expressions for L 1 and L 2 giv en in (3.30) w e arrive at the condition (3.29). It is known that this eight-dimensional Lie algebra of p oint symmetries is isomorphic to sl (3 , R ). This completes the pro of. Example 1 . W e provide an example of an equation from (2.5) that illustrates that all dimen- sions of p ( c ) listed in Theorem 3.1 are p ossible. Consider the oscillator y xx +  1 y + µ − 1 y 2 + ν 3 y 3  y 3 x +  3 + µ − 2 y + ν 3 y 2  y 2 x + 3 y y x + y 2 = 0 , (3.39) First, it is not difficult to c heck that co efficients of (3.39) satisfy (3.17). Then, using (3.8) we find that B = 3 µy + ν . Substituting B and g into (3.26) we obtain that it v anishes if and only if µν = 0. Consequently , for arbitrary non-zero v alues of the parameters µ and ν w e hav e that dim p ( c ) | (3.39) = 1. It is also clear that B y = 3 µ and, hence, if µ = 0, then dim p ( c ) | (3.39) = 8. Th us, further, we assume that µ  = 0 and ν = 0. Substituting the expressions for B and g into (3.26) w e obtain that dim p ( c ) | (3.39) = 2 if µ  = 3 / 4. Using relations (3.27) we obtain that dim p ( c ) | (3.39) = 3 if and only if µ = 3 / 4. 4 Metrisabilit y of cubic oscillators In this section w e classify all metrisable cubic oscillators that satisfy condition (3.17). W e also demonstrate that all metrics describ ed in Theorem 3 of [16] are either Darb oux-sup erin tegrable or flat. W e show ho w to obtain non-trivial sup erin tegrable metrics for metrisable equations from (2.3) under correlation (3.17). Theorem 4.1. Supp ose that c orr elation (3.17) holds and g  = 0 . Then e quation (2.5) is metrisable if and only if one of the fol lowing c onditions holds: 1) ψ 1 ,x = 0 ; 2) c orr elation (3.27) is fulfil le d or dim p ( g ) = 3 ; 3) c orr elation (3.29) is fulfil le d or dim p ( g ) = 8 . Pr o of. In order to pro of this statemen t we find compatibilit y conditions for Lioiville system (2.7) for (2.5). This is done by excluding ψ 2 and ψ 3 from (2.7) via ψ 2 = 3 ψ 1 ,x + 2 f ψ 1 6 g , ψ 3 = 9 ψ 1 ,xx + 3 f ψ 1 ,x + 9 g ψ 1 ,y − 2 f 2 ψ 1 + 12 hg ψ 1 18 g 2 , (4.1) and obtaining a system of t wo linear partial differential equations for ψ 1 , whic h has the form 27 ψ 1 ,xxx + 81 g ψ 1 ,xy − 27 f ψ 1 ,xx − 18  f 2 − 3 g h − 3 g y  ψ 1 ,x +  8 f 3 − 36 f g h + 108 g 2 k − 36 f g y + 36 g f y  ψ 1 = 0 , (4.2) 12 27 g ψ 1 ,xxy + 9 f ψ 1 ,xy g + 27 g 2 ψ 1 ,y y − 18 ( g h + 3 g y ) ψ 1 ,xx − 3  2 f g h − 18 g 2 k + 6 f g y − 3 g f y  ψ 1 ,x − 3  2 f 2 − 6 g h + 9 g y  g ψ 1 ,y + 4  f 2 g h + 9 f g 2 k − 6 g 2 h 2 + 3 f 2 g y − 3 f g f y + 9 g 2 h y − 9 g hg y  ψ 1 = 0 (4.3) Belo w w e obtain in tegrability conditions for (4.2), (4.3) as an o v erdetermined system of tw o equa- tions for the function ψ 1 . First, w e exclude the function k from (4.2)-(4.3) via (3.17). Then, in order to simplify the computations, we use the definition of B giv en in (3.8) to exclude the function h from (4.2)-(4.3). This yields 3 ψ 1 ,xxx + 9 g ψ 1 ,xy − 3 f ψ 1 ,xx − 2 (6 g y + B ) ψ 1 ,x = 0 , (4.4) 81 g ψ 1 ,xxy + 81 g 2 ψ 1 ,y y + 27 f g ψ 1 ,xy − 18  f 2 − B + 6 g y  ψ 1 ,xx − 9 g (15 g y + 2 B ) ψ 1 ,y − 3 (4 B f + 12 f g y + 9 g f y ) ψ 1 ,x − 4  2 B 2 − 6 B g y + 9 g B y + 27 g g y y − 36 g 2 y  ψ 1 = 0 . (4.5) T o obtain integrabilit y conditions of (4.4)-(4.5) we follow the classical approac h b y Liou- ville [27]. Cross-differen tiating equations (4.4)-(4.5), w e find an expression for ψ 1 ,xy y . Then, differen tiating ψ 1 ,xy y b y x and equation (4.4) b y y w e obtain the expression for ψ 1 ,y y y . Conse- quen tly , w e ha ve expressions for all third order partial deriv ativ es through the deriv ativ es of lo wer order. Finally , w e compute the difference b et ween ψ 1 ,xy y y and ψ 1 ,y y y x and using all third order deriv ativ es we obtain the correlation 45 g B y ψ 1 ,xy − 15 f B y ψ 1 ,xx + (9 g B y y − 2 B B y − 51 g y B y ) ψ 1 ,x = 0 . (4.6) Since g  = 0 further computations split into tw o cases: B y  = 0 and B y = 0. Supp ose that B y  = 0. Then w e solv e (4.6) for ψ 1 ,xy and differen tiate the former by x . As a result, we find an expression for ψ 1 ,y y . If we differentiate (4.6) by y and eliminate ψ 1 ,xy y , ψ 1 ,xxy and ψ 1 ,xy w e obtain ψ 1 ,x  45 g 2 B y B y y y − 54 g 2 B 2 y y − 9 g B y ( B − 3 g y ) B y y +  4 B 2 − 9 B g y − 15 g B y + 45 g g y y − 9 g 2 y  B 2 y  = 0 (4.7) Th us, further computations split into tw o cases: ψ 1 ,x = 0 and ψ 1 ,x  = 0. Supp ose that ψ 1 ,x = 0. Then the equation (4.4) is fulfilled and from (4.5) we find that ψ 1 is a solution of a linear homogeneous equation 81 g 2 ψ 1 ,y y − 9 g (15 g y + 2 B ) ψ 1 ,y − 4  2 B 2 − 6 B g y + 9 g B y + 27 g g y y − 36 g 2 y  ψ 1 = 0 . (4.8) Notice that if ψ 1 ,x = 0 from (4.1) it follows that ψ 2 ,x = ψ 3 ,x = 0. Thus, we see that equations from (2.5) that satisfy (3.17) are metrisable without an y additional conditions on the functions f , g , h and k , if the corresp onding metric do not dep end on x . Let us con tin ue with the case of ψ 1 ,x  = 0. One more differentiation of (4.6) by x yields correlation (3.27). The first deriv ativ e of (3.27) is presented in (3.38). If w e assume that C 3  = 0 in (3.38) and eliminate B y y y and B 2 y y via (3.38) and (3.27), resp ectively , w e obtain that (4.7) v anish. F urther differentiating of (4.6) does not lead to any new compatibility conditions. If we assume that C 3 = 0 (see (3.35) for the definition) then w e obtain that (3.27) and (4.7) v anish simultaneously if and only if B y = 0 and this case is considered b elo w. 13 Supp ose that B y = 0. Then (4.6) v anishes and w e ha v e the expressions of all third order deriv ativ es of ψ 1 via lo wer order deriv atives. Cross-differen tiation of this third-order deriv ativ e do es not result in an y new compatibility conditions. Consequen tly , if B y = 0 an equation from (2.5) under conditions (3.17) is metrisable. This completes the pro of. Corollary 4.1. Metrics that c orr esp ond to (2.3) when c ondition (3.17) is fulfil le d and with ψ 1 ,x  = 0 , which ar e c onsider e d in The or em 3 of [16], ar e either Darb oux-sup erinte gr able or flat. Pr o of. In [1] it was demonstrated that if dim p ( g ) = 3 then g is Darboux-sup erin tegrable, i.e. g has 4 linearly indep enden t quadratic, with resp ect to momen ta, first integrals. These metrics were classified by Ko enigs [40]. If dim p ( g ) = 8, then the corresp onding pro jective equation is point equiv alen t to y xx = 0 and, th us, g has a constan t curv ature. Such metrics has three, linearly indep enden t, linear in momen ta first integrals. No w we demonstrate that autonomous pro jectiv e structures considered in [16] satisfy condition (3.17). T o this aim we use the notation of [16]. The integrabilit y conditions for the cubic oscillator y xx = A 3 y 3 x + A 2 y 2 x + A 1 y x + A 0 . (4.9) presen ted in formula (2.18) in [16] hav e the form A 1 = − 3 A 0 v , A 2 = 3 A 0 v 2 + 3 v y 2 v − w y 2 w , A 3 = − A 0 v 3 + 1 2 v  w y w − v y v  . (4.10) Here v and w , v w  = 0 are axillary functions. In [16] the metrisabilit y of (4.9) when (4.10) holds is studied. Notice also that in [16] it is assumed that A 0 A 3  = 0. Ho wev er, the requiremen t A 3  = 0 can b e omitted. Excluding the functions v and w from the first t wo equations of (4.10) and substituting the results in to the third one we obtain A 3 = 9 A 2 A 1 A 0 − 2 A 3 1 − 9 A 0 ,y A 1 + 9 A 1 ,y A 0 27 A 2 0 . (4.11) Since A 0  = 0 from (4.11) w e get 27 A 2 0 A 3 − 9 A 2 A 1 A 0 + 2 A 3 1 + 9 A 0 ,y A 1 − 9 A 1 ,y A 0 = 0 . (4.12) No w comparting (2.5) and (4.9) w e see that A 0 = − g , A 1 = − f , A 2 = − h and A 3 = − k . Consequen tly , (4.10), (4.11) and (4.12) are equiv alen t to 27 k g 2 − 9 hf g + 2 f 3 + 9 g f y − 9 f g y = 0 . (4.13) T aking into accoun t definition (3.8) of the function A we obtain that conditions (4.10) are equiv alen t to linearizabilit y condition (3.17). Consequen tly , w e see that, in fact, in Theorem 3 of [16] the metrisability of (2.5) under condition (3.17) is studied. Moreov er, in [16] it is assumed that ψ 1 ,x  = 0. Therefore, due to the first part of this corollary , the corresp onding metrics are either Darb oux-sup erin tegrable or flat. F urthermore, this means that all metrics presen ted in [16] ha ve p olynomial first integrals, and the rational first in tegrals provided in [16] are fractions of these p olynomials. Finally , let use remark that a particular case of condition (3.17) for k = 0 and constant h was considered in [41]. In general, correlation (3.17) can b e considered as an in tegrabilit y conditions for the family of the Abel differential equations z y = g z 3 + f z 2 + hz + k , y x = z − 1 (see, [42] 4.10(c)). This completes the pro of. 14 Corollary 4.2. Al l metrics that c orr esp ond to (2.5) under c ondition (3.17) ar e sup erinte gr able. Their ge o desics ar e forme d by the curves ( x, x ( y )) =   x, c 2 − y Z 0 f ( ω ) 3 g ( ω ) − s C ′ ( ω ) g ( ω )[ c 1 − 2 C ( ξ )] ! dω   , (4.14) wher e c 1 and c 2 ar e arbitr ary c onstants. Pr o of. If ψ 1 ,x = 0, then from the relations (4.1) it follows that the metric g and Hamiltonian H in (2.2) do not explicitly dep end on x . Therefore, there exists a linear first in tegral L = p 1 . On the other hand, if correlation (3.17) holds, then (2.5) admits in v arian t (3.24). If w e lift this inv arian t via y x = u = H p 2 /H p 1 , we obtain a first in tegral of Hamiltonian system (2.2) that explicitly dep ends on x . Clearly , this yields that H , L and lifted (3.24) are functionally independent (see also Prop osition 1 in [15]). If ψ 1 ,x  = 0 and both conditions (3.17) and (3.27) are fulfilled, then g is Darb oux sup erinte- grable. In the case of the flat metrics that are describ ed b y (3.29) it is known [2] that they are sup erin tegrable with t wo linear with resp ect to momenta first integrals. Lastly , supp ose that R = c 1 and R = c 2 , where c 1 and c 2 are arbitrary constants. Then, from (3.23) and (3.24) w e find (4.14). This completes the pro of. Corollary 4.3. F or any e quation fr om (2.5) whose c o efficients satisfy (3.17) ther e is the fol lowing solution of the Liouvil le system ψ 1 = ( c 3 + c 4 C )  g C y  2 / 3 , ψ 2 = ( c 3 + c 4 C ) f 3( g C 2 y ) 1 / 3 , ψ 3 = 9 c 4 g C y + f 2 ( c 3 + c 4 C ) 18( g 2 C y ) 2 / 3 , g C y  = 0 , (4.15) wher e C is given in (3.21) and c 3 and c 4  = 0 ar e arbitr ary c onstants. Mor e over, r ational first inte gr al (3.23) is a pr oje ction of a c ombination of the Hamiltonian that c orr esp onds to (4.15) and its line ar first inte gr al L = p 1 . Pr o of. Suppose that in (4.4)-(4.5) ψ 1 ,x = 0. Then, (4.4) v anish iden tically and (4.5) is a linear homogeneous second-order equation for ψ 1 . Excluding B via (3.20), its general solution can b e easily found and is presen ted in (4.15). Then, with the help of (4.1) w e find the rest of the expressions from (4.15). One can find that for (4.15) w e ha ve that ∆ = ( c 3 + c 4 C )( g /C y ) 1 / 3 ( c 4 / 2). Since g C y  = 0 w e see that w e need to require c 4  = 0 so that the corresp onding metric g do not degenerate. With the help of the expressions (4.15), one can find the metric tensor g ij , its inv erse and the Hamiltonian for the geo desic flo w H = c 4 C + c 3 72 g C y  [(2 C f 2 + 9 g C y ) c 4 + 2 f 2 c 3 ] p 2 1 − 12 g f ( c 4 C + c 3 ) p 1 p 2 + 18 g 2 ( c 4 C + c 3 ) p 2 2  (4.16) Clearly , that this Hamiltonian also admits L = p 1 as a first in tegral. Then, w e can pro ject the expression H /p 2 1 on the ( x, y ) plane via y x = u = H p 2 /H p 1 . As a consequence, we get R = 9 c 3 4 g C y u 2 16( f u + 3 g ) 2 + c 3 4 8 C + c 3 c 2 4 8 . (4.17) 15 Without lo os of generalit y , setting c 2 = 2 4 / 3 , w e obtain (3.23). This completes the pro of. Finally , we provide several examples of metrisable equations from (2.5) that correspond to sup erin tegrable metrics. Example 2. Let us consider the follo wing oscillator from (2.3) that satisfy condition (3.26) y xx + (81 y 6 + 27 y 3 − 7) y 3 x + 243 y 6 + 54 y 3 − 4 3 y y 2 x + 3 y (9 y 3 + 1) y x + 3 y 3 = 0 . (4.18) One can also sho w that conditions (3.27) and (3.29) cannot b e fulfilled for (4.18). With the help of relations (3.23) and (3.24) we find autonomous R = y − 2 / 3  1 − 9 y 2 u 2 [3 y (9 y 3 + 1) u + 9 y 3 ] 2  , (4.19) and non-autonomous R = x + 3 y 2 (9 u y 3 + 3 y 2 + u ) 3   729 y 9 + 81 y 6 + 27 y 3 + 7  y u 3 +  9 y 3 + 1   81 y 6 − 9 y 3 + 2  u 2 + 27 y 5  9 y 3 − 1  u + 3 y 4  9 y 3 − 2   . (4.20) first in tegrals of (4.18), resp ectiv ely . Equation (4.18) is metrisable and the corresponding Hamiltonian for geo desics is H = y − 4 / 3  3(9 y 3 + 2) y p 2 1 2 − (9 y 3 − 1) p 1 p 2 + 3 y 2 p 2 2 2  . (4.21) This Hamiltonian system is sup erintegrable with 3 functionally independent integrals H , L = p 1 and the lift of (4.20), whic h is T = 1 2  2 x − 3 y 2  18 y 3 + 5   18 y 3 + 1  p 3 1 + 3 y  162 y 6 + 36 y 3 + 1  p 2 p 2 1 − 18 y 3  9 y 3 + 1  p 2 2 p 1 + 18 y 5 p 3 2 . (4.22) Notice that dim p ( g ) = 2 for (4.21) since condition (3.26) is met. Pro jective equation (4.18) pro vides an example of non-trivial metrisable equation with a rational first in tegral that satisfy condition (3.26). Example 1 con tinued. F or arbitrary v alues of µ and ν the first in tegral of (3.39) is expresses in terms of the gamma function and the relativ e non-autonomous first in tegral is difficult to obtain. Let us discuss sev eral particular cases of (3.39), when b oth autonomous and non-autonomous first in tegrals can b e presented in the explicit form. Supp ose, that ν = 0 and µ = 1. Then, (3.39) p ossesses the first integral R = y 2  u 2 y 2 ( u + y ) 2 + 2 y  , (4.23) and in v ariant R = ( y e) √ (2 y +1) u 2 +4 uy 2 +2 y 3 u + y ( p (2 y + 1) u 2 + 4 uy 2 + 2 y 3 + u ) 2 2 u 2 y + 4 uy 2 + 2 y 3 . (4.24) 16 Equation (3.39) is metrisable for arbitrary v alues of µ and ν . A t µ = 1 and ν = 0 the corresponding Hamiltonian for geo desics is H = y (2 y − 1)(2 p 2 1 − 2(2 y − 1) p 1 p 2 + (2 y − 1) y p 2 2 ) . (4.25) This Hamiltonion has the linear first in tegral L = p 1 . One can demonstrate that lifted first in tegral (4.23) is a function of L and H , while (4.25) is functionally indep endent of L and H . Therefore, (4.25) is sup erin tegrable with linear and a transcendental first integrals. P arameters dim p ( c ) 1 β 3 = α 3 ( β 0 + 1) , β 2 = α 1 (2 β 0 + 1) / 2 , β 1 = 2 β 0 ( β 0 − 1) 2 + α 2 3 + α 2 1  = 0 , 1 , α 2 3 + α 2 1  = 0, 8 , α 3 = α 1 = 0 2 β 3 = β 2 = α 1 = 0 , β 1 = − 4 , β 0 = 2 1 , α 3  = 0, 8 , α 3 = 0 3 β 3 = β 2 = α 1 = 0 , β 1 = 2 , β 0 = − 1 1 , α 3  = 0, 8 , α 3 = 0 4 β 3 = β 2 = α 1 = 0 , β 1 = − 1 ± 3 √ 3 , β 0 = − 4, 1 , α 3  = 0, 8 , α 3 = 0 5 β 3 = 0 , β 1 = − 5 / 2 , β 0 = − 1 , α 1 = 5 β 2 / 2 , α 3 = β 2 2 1 , β 2  = 0, 8 , β 2 = 0 6 β 3 = 0 , β 1 = 2 , β 0 = − 1 , α 1 = − 2 β 2 1 , α 2 3 + β 2 2  = 0, 8 , α 3 = β 2 = 0 T able 1: V alues of the parameters that corresp ond to metrisable cases of system (4.26) Example 3 W e consider a v arian t of the cubic Kolmogorov system y x = y ( α 0 − α 1 y − α 2 z − α 3 y 2 ) , z x = z ( β 0 − β 1 z − β 2 y − β 3 y 2 − β 4 z 2 ) . (4.26) Here α j , j = 0 , 1 , 2 , 3 and β k , k = 0 , 1 , 2 , 3 , 4 are arbitrary parameters. W e assume that α 0 α 2 β 4  = 0. Th us, without loss of generalit y , w e set α 0 = α 2 = β 4 = 1. Then, w e can exclude z via the first equation from (4.26) and substitute the result in to the second one. As a consequence, we obtain an oscillator from (2.5), whic h is y xx + y 3 x y 2 +  3 a 3 y + 3 a 1 − b 1 + 4 y  y 2 x +  3 a 2 3 y 4 + 6 a 1 a 3 y 3 + +(3 a 2 1 − 2 a 3 b 1 − 4 a 3 + b 3 ) y 2 − (2 a 1 b 1 + 5 a 1 − b 2 ) y − b 0 + 2 b 1 + 3  y x + + y  a 3 y 2 + a 1 y − 1  a 2 3 y 4 + 2 a 3 a 1 y 3 + ( a 2 1 − a 3 b 1 − 2 a 3 + b 3 ) y 2 + +( b 2 − a 1 b 1 − 2 a 1 ) y − b 0 + b 1 + 1  = 0 . (4.27) No w w e substitute the co efficien ts of (4.27) in to (3.17) and find the v alues of the parameters when this correlation is fulfilled. The results are presented in T able 1. Notice that w e do not consider complex v alues of the parameters and do not show cases of equations with dim p ( g ) = 8 for an y v alues of the parameters. F or any v alue of the parameters an equation from (4.27) will b e metrisable and ha ve a rational first in tegral. The existence of this rational first integral follo ws from in tegrability of the corresp onding geo desic flow with a linear first integral L = p 1 . 17 Ho wev er, metrisable equations from (4.27) also hav e an inv arian t given by (3.24) and, hence, explicit representation for geo desics (4.14). F or example, let us consider the case 8 from T able 1. F or a sake of simplicity , w e also assume that β 2 = 0 and α 3 = − γ 2 . As a result, we find that the geo desic flo w H = − γ 2  p 2 1 2 − γ 2 y 3 p 1 p 2 + γ 2 y 4 ( A 2 y 2 − 1) 2 p 2 2  , (4.28) is sup erin tegrable with tw o additional firs in tegrals L = p 1 , T = (2 p 1 − ( γ 2 y 3 − y ) p 2 ) y 3 p 2 e 2 x − 2 p 1 ( γ 2 y 3 p 2 − yp 2 − p 1 ) y 3 γ 2 ( γ 2 y 3 p 2 − yp 2 − 2 p 1 ) p 2 . (4.29) There are other examples of explicitly superintegrable geo desic flo ws among cases listed in T able 1. 5 In tegrable conformal metrics In the previous Section the metrisability of equations from (2.5) than can be linearized via (3.2) is studied only in one case, namely when correlation (3.17) holds. Finding metrisable equations that satisfy either condition (3.13) or (3.15) for a generic metric g is computationally difficult. Ho wev er, this can b e ov ercome for sp ecial cases of the metric g . First, we assume that g 12 = g 21 = 0, which implies that ψ 2 = 0. Then w e in addition supp ose that g 11 = g 22 , which yields ψ 1 = ψ 3 . The former case of g corresp ond to the orthogonal metrics, while the latter one is called conformal metrics [43]. Theorem 5.1. Equation (2.5) is metrisable in the c ase of ψ 2 = 0 if and only if either f k y = − 2 g k 2 + 2 f hk , f y = 0 , f k  = 0 (5.1) or k = f = 0 . (5.2) The solution of the Liouvil le system (2.7) is either ψ 1 = ϵ k ψ 3 , ψ 3 = c 5 exp  2 3  Z hdy − ϵx  , f = ϵ  = 0 , c 5 k  = 0 , (5.3) or ψ 1 = e − 4 3 R hdy  2 c 6 Z g e 2 R hdy dy + c 7  , ψ 3 = c 6 e 2 3 R hdy , (5.4) r esp e ctively. Her e c 5 , c 6 and c 7 ar e arbitr ary c onstants such that c 5 c 6  = 0 . In the c ase of the c onformal metric g , i.e. ψ 1 = ψ 3 and ψ 2 = 0 , we have that (2.5) is metrisable if and only if k = f , h = g , f y = 0 . (5.5) The solution of the Liouvil le system is ψ 1 = ψ 3 = c 5 exp  2 3  Z g dy − ϵx  , f = k = ϵ  = 0 , c 5  = 0 . (5.6) 18 Pr o of. First, we substitute that a 3 = k , a 2 = h , a 1 = f , a 0 = g and ψ 2 = 0 in to (2.7). As a result, w e hav e four equations for t wo functions ψ 1  = 0 and ψ 3  = 0, whic h are 3 ψ 1 ,x + 2 f ψ 1 =0 , 3 ψ 3 ,y − 2 hψ 3 = 0 , 3 ψ 1 ,y + 4 hψ 1 − 6 g ψ 3 =0 , 3 ψ 3 ,x + 6 k ψ 1 − 4 f ψ 3 = 0 . (5.7) Cross differentiating the first and the third and the second and the fourth equations from (5.7) we get 12 f g ψ 3 − 12 g k ψ 1 + 2 f y ψ 1 = 0 , 12 hk ψ 1 − 12 k g ψ 3 − 6 k y ψ 1 + 4 f y ψ 3 = 0 . (5.8) Recall, that throughout this w ork w e assume that g  = 0. Then, from the first equation of (5.8) w e see that we need to consider tw o cases separately: f  = 0 and f = 0. Supp ose that f  = 0. Then we exclude ψ 3 from the first equation from (5.7) and then substitute the result in to the second one. Recall that if ψ 2 = 0, then ∆ = ψ 1 ψ 3  = 0. As a consequence, taking in to account that ψ 1 f g  = 0, we get ψ 3 = (6 g k − f y ) ψ 1 6 f g , 9 g (2 hk f − 2 g k 2 + k f y − f k y ) − f 2 y = 0 . (5.9) Then, substituting the expression for ψ 3 along with the expression for ψ 1 ,x , obtained from the first equation of (5.7), to the last equation of (5.7) w e obtain that f y = 0. As a result, we immediately get compatibilit y conditions (5.1) and the expression for ψ 3 from (5.3). Supp ose that f = 0. Then, from the first equation of (5.8) w e find that k = 0 and the second one is satisfied automatically . Substituting f = k = 0 into (5.7) w e find the expressions for ψ 1 and ψ 3 giv en in (5.4). Finally , w e consider the case of the conformal metrics, i.e. we assume that ψ 3 = ψ 1  = 0. As a consequence, from (5.7) we obtain that g = h and f = k . Then, comparing ψ 1 ,xy and ψ 1 ,y x w e obtain that f y = 0. The expression for ψ 1 can b e directly found from the fist line of (5.7). Notice also that the ab o ve metrics hav e infinitesimal homotet y ∂ x . This completes the pro of. No w let us find in tersections b etw een families of linearizable equations provided in Prop osition 3.2 and metrisable oscillators giv en in Theorem 5.1. First, we consider the case of conformal metric g and linearizabilit y condition (3.15): Theorem 5.2. The metric g of the form ds 2 = exp  2  x − Z g dy   dx 2 + dy 2  , (5.10) with the ge o desic flow with the Hamiltonian H = 1 2 exp  2  Z g dy − x   p 2 1 + p 2 2  , (5.11) is inte gr able with the tr ansc endental first inte gr al T = K 1 − i √ 3 1 K 1+ i √ 3 2 K − 2 3 K 4 , (5.12) or in the r e al form T = exp ( 2 √ 3 arctan √ 3(( l − 1) p 2 − 3 g p 1 ) K 3 !) K 5 K 2 3 , (5.13) 19 if and only if the function g  = 0 is a solution of the e quation B A − g A y = 9 g g y y − 27 g 2 y −  9 g 2 − 15  g y + 2  9 g 2 + 1   3 g 2 − 1  = 0 . (5.14) Her e we use the notations A = 18 g 2 + 2 − 9 g y , B = 3 g y + 3 g 2 − 1 , l 3 = A, (5.15) and K j , j = 1 , 2 , 3 , 4 ar e the r elative Kil ling ve ctors that given by K 1 , 2 = 12 g p 1 + (1 ∓ √ 3 i )(1 − 2 l ± √ 3 i ) p 2 , K 3 = 3 g p 1 + ( l + 1) p 2 , K 4 = exp  2 3 Z l 2 g dy  , K 5 = K 1 K 2 = 9 g 2 p 2 1 − 3( l − 2) g p 1 p 2 + 16( l 2 − l + 1) p 2 2 . (5.16) Pr o of. F rom Theorem 5.1 it follows that an equation from (2.5) is metrisable in the case of the conformal metric if and only if it has the form y xx + ϵy 3 x + g y 2 x + ϵy x + g = 0 , ϵg  = 0 . (5.17) Notice that, without loss of generalit y , we can set ϵ = 1. Then, w e apply to (5.17) integrabilit y condition (3.15) from Proposition 3.2. As a result, we obtain that (5.17) can b e transformed to linear equation (3.1) with (3.14) via (3.2) if and only if correlation (5.14) is fulfilled. F rom Prop osition 3.2 it also follo ws that linearizabilit y is equiv alen t to the existence of the first in tegral (3.19). Under (3.14) the expression for (3.19) is R = exp  2 3  Z l 2 g dy  h (1 + √ 3 i )(1 − √ 3 i − 2 l ) u + 12 g i 1+ √ 3 i × h (1 − √ 3 i )(1 + √ 3 i − 2 l ) u + 12 g i 1 − √ 3 i [(1 + l ) y x + 3 g ] 2 . (5.18) Then w e lift (5.18) to (5.12) via u = y x = H p 2 /H p 1 . The real v arian t of in tegral (5.18) has the form R = exp ( 2 3  Z l 2 g dy  − 2 √ 3 arctan √ 3 ( ϵy x − l y x + 3 g ) l y x + ϵy x + 3 g !) × (9 g 2 + 3 y x (2 ϵ − l ) g + y 2 x ( ϵ 2 − ϵl + l 2 )) (( l + ϵ ) y x + 3 g ) 2 (5.19) In the same w ay it can b e lifted into (5.13). One cans see that the first in tegral (5.18) is formed b y the inv ariants of (2.5), that are linear in the first deriv ative. Consequently , using Prop osition 2.1 one can find the corresp onding relative Killing v ectors. This completes the pro of. No w we pro ceed with the most general linearizability criterion from Prop osition 3.2. Theorem 5.3. The metric g of the form ds 2 = exp  2  x − Z g dy   dx 2 + dy 2  , (5.20) 20 with the ge o desic flow with the Hamiltonian H = 1 2 exp  2  Z g dy − x   p 2 1 + p 2 2  , (5.21) is inte gr able with the first inte gr al T = K ρ + β 6 K ρ − β 7 K − 2 ρ 8 K 9 , (5.22) if and only if the function g  = 0 is a solution of the e quation β 2 (2 β 2 − 9 α ) 2 ( g A y − B A ) 3 − A 5 ( β 2 − 3 α ) 3 = 0 , (5.23) such that (2 β 2 − 9 α )( β 2 − 3 α ) A  = 0 . Her e the functions A , B and l ar e pr ovide d in (5.15) and ρ = p β 2 − 4 α , δ = 2 β 2 − 9 α . (5.24) If, in addition, the c orr elation 4 α = β 2 (1 − r 2 ) , r ∈ Q , r  = ± 1 , r  = ± 1 3 , (5.25) holds, then metric (5.20) is inte gr able with a r ational first inte gr al of an arbitr ary de gr e e given by T r = K p + q 1 K p − q 2 K − 2 p 3 K q β 4 . (5.26) The r elative Kil ling tensors K j , j = 6 , 7 , 8 , 9 have the form K 1 , 2 = (3 ρ ∓ β ) l p 2 ∓ 2( δ β ) 1 3 (3 g p 1 + ϵp 2 ) , K 3 = β 2 3 l p 2 − δ 1 3 (3 g p 1 + ϵp 2 ) , K 4 = exp  2 αρ ( δ β ) − 2 3 Z l 2 g − 1 dy  . (5.27) Pr o of. The pro of is similar to those of Theorem 5.2. First, w e observe that the pro jectiv e equation that corresp ond to (5.21) is (5.17) and set ϵ = 1. Then, w e apply integrabilit y criterion (3.17) to (5.17). Consequen tly , we get that (5.17) is linearizable and has first integral (3.19) if and only if (5.23) is fulfilled. Then, after some simplifications, from (3.19) w e obtain T = exp ( 2 αρ R l 2 g dy ( δ β ) 2 3 ) h β 2 3 l y x − δ 1 3 ( ϵy x + 3 g ) i − 2 ρ × h ( β + 3 ρ ) ly x + 2( β δ ) 1 3 ( y x + 3 g ) i ρ − β h (3 r − β ) l y x − 2( β δ ) 1 3 ( y x + 3 g ) i ρ + β . (5.28) This in tegral can b e lifted to (5.22). Finally , in order to obtain a rational first integral from (5.22) w e substitute (5.25) into it. Notice that we exclude the cases of r 2 = 1 and r 2 = 1 / 9 since in the former α = 0 and in the latter 2 β 2 − 9 α = 0. The relativ e Killing tensors K j , j = 6 , 7 , 8 , 9 can b e obtained via formulas (2.9) from Prop osition 2.1 from the multipliers of (5.28), which are in v arian ts linear in u of the vector field asso ciated with (2.5). This completes the pro of. 21 Notice that w e do not provide explicit expressions for the cofactors of the relative Killing vectors giv en in Theorems 5.2 and 5.3 since they hav e a cum b ersome form. Let us connect and generalize the results of Theorems 5.2 and 5.3 b y in tro ducing the concept of generalized Darb oux functions and generalized Darb oux integrabilit y for both pro jective equation (2.3) and Hamiltonian system (2.2). Recall that, in the classical theory of in tegrability of p olyno- mial finite-dimensional vector fields, the Darb oux functions are defined as f s 1 1 . . . f s r r exp n f r +1 f r +2 o , where g , f , f s 1 1 . . . f s r r ∈ C [ x ], s 1 , . . . , s r ∈ C and x ∈ R n [20–22]. A finite-dimensional v ector field has a first integral whic h is a Darb oux function, if and only if the multipliers of this Darb oux func- tion are p olynomial and/or exp onential inv ariants of this v ector field, whose cofactors are linearly dep enden t [20–22]. It is said that a p olynomial n -dimensional vector field is Darb oux integrable if it has n − 1 functionally indep endent first integrals that are Darb oux functions. On the other hand, one can see that the v ector fields X Γ and X H are p olynomial only in the v elo cities and momenta, resp ectively , while b oth vector fields X p and X are p olynomial only in the first deriv ativ e u = y x . F urthermore, from Prop osition 3.2 and Theorems 5.2 and 5.3, w e see that these v ector fields can hav e first in tegrals which are Darb oux functions of in v arian ts that are p olynomial only in the abov e subsets of the phase v ariables. Th us, it is natural to relax the requiremen t that b oth the v ector fields and the inv ariants are p olynomials and assume that they are p olynomial only in a subset of the phase v ariables. A similar concept for t w o-dimensional v ector fields w as considered in [23–25]. This allo ws us to in tro duce the definitions of a gener alize d Darb oux function and gener alize d Darb oux inte gr ability for b oth pro jective and Hamiltonian geodesic v ector fields. First, w e consider the pro jective vector fields X p and their inv arian ts R that b elong to C ∞ ( M )[ u ] with the cofactors l also b elonging to C ∞ ( M )[ u ]. W e will call the Darb oux functions of such in- v arian ts generalized Darboux functions. Moreov er, we sa y that the v ector field X p is generalized Darb oux in tegrable if it has tw o functionally indep endent first in tegrals whic h are generalized Dar- b oux functions. It is a straightforw ard computation to verify that the v ector field X p is generalized Darb oux in tegrable if and only if the cofactors of the m ultipliers of the corresp onding generalized Darb oux function are linearly dep enden t (see, e.g. [25]). Second, in a similar wa y , we can introduce the definition of a generalized Darb oux function and generalized Darb oux integrabilit y for Hamiltonian system (2.2) using relativ e Killing tensors. Indeed, b y definition, K and the resp ective cofactors b elong to C ∞ ( M )[ p 1 , p 2 ] (see, [12–14]). W e will call the Darb oux functions of the relativ e Killing tensors generalized Darb oux functions. W e will call a Hamiltonian system integrable in the generalized Darb oux sense if it has a first integrals that is a generalized Darb oux function, is in inv olution with the Hamiltonian and is functionally indep enden t of it. One can see that b oth the linearizable cases of (2.5) and Hamiltonian (5.21) under either condition (5.14) or (5.23) provide examples of generalized Darb oux in tegrable finite-dimensional v ector fields and Hamiltonian systems, corresp ondingly . F urther details and algorithms for the construction of generalized Darb oux first integrals for b oth the pro jectiv e equation and Hamilto- nian geodesic flow will b e dev elop ed elsewhere. Note also that in [25] this type of in tegrals and in v ariants was systematically considered for t w o-dimensional vector fields. Prop osition 5.1. Conditions (5.14) and (5.23) ar e inte gr able as e quations for the function g . Pr o of. W e b egin with condition (5.14). If we integrate it as an equation for A we obtain R = Ag − 3 exp  Z  ϵ 2 g − 3 g  dy  . (5.29) 22 If w e differentiate (5.29) with resp ect to y , w e obtain (5.14). On the other hand, w e can directly consider in tegrability of (5.14) as an equation for g . One can demonstrate, for example using the approach from [25], that (5.14) as a dynamical system has 3 first order with resp ect to g y in v ariants curves. T aking in to account that g itself is also an in v ariant curve we obtain the following integrating factor for (5.14) M = g  9 g 2 y ϵ 2 − 6 ϵ 2 ( ϵ 2 − 3 g 2 ) g y + ( ϵ 2 + 9 g 2 ) ( ϵ 2 − 3 g 2 ) 2  (9 g y − 2 ϵ 2 − 18 g 2 ) 1 3 . (5.30) Notice that these in v arian t curv es are in the denominator of (5.30). No w we pro ceed with the condition (5.23). This is a second-order third-degree differen tial equation for g , i.e. it is cubic in g y y . Th us, (5.23) can b e considered as a multiplication of three second-order first-degree differen tial equations for g y y . Each of these three equations has a first in tegral, which can b e presented in a compact form via the function C as follows R 1 = g A − 2 / 3 C y + 2( β 2 − 3 α ) C 3 β 2 / 3 (2 β 2 − 9 α ) 2 / 3 , R 2 , 3 = g A − 2 / 3 C y + ( ± i √ 3 − 1)( β 2 − 3 α ) C 3 β 2 / 3 (2 β 2 − 9 α ) 2 / 3 (5.31) This completes the pro of. Prop osition 5.2. In the generic c ases dim p ( g ) = 1 for metrics (5.10) define d by (5.14) and (5.23) . Pr o of. The pro of is straightforw ard. First, observ e that for (2.5) the pro jective Lie algebra is at least one-dimensional. Second, w e compute relative differen tial in v ariants J j , j = 0 , 1 , 2 , 3 , 4 for equations from (2.5) of the form (5.5) and that satisfy either (5.14) or (5.23). Third, we find that J 0  = 0 and basis algebraic inv arian ts I j , j = 1 , 2 , 3 , 4 are not constant. Finally , according to [34] the pro jectiv e Lie algebra is t w o=dimensional if and only if all basis inv ariants are constan t. This completes the pro of. 6 Conclusion In this work w e hav e considered metrisability of the autonomous case of the pro jective con- nection. W e hav e constructed three families of (sup er)in tegrable t wo-dimensional metrics that are parameterised b y arbitrary functions. In the first case, we ha v e obtained a family of sup erinte- grable metrics with an explicit expression for the unparameterized geo desics. In the other t wo cases we hav e constructed families of integrable metrics with transcenden tal first integrals, which at some v alues of the parameters degenerate into a rational one with an arbitrary degree. W e ha ve studied the dimensions of the pro jectiv e Lie algebras of obtained metrics. W e show that all constructed metrics generically hav e one-dimensional pro jectiv e Lie algebra. W e ha ve found particular instances, when this Lie algebra is 2, 3 and 8 dimensional. This has allo wed us to demonstrate when obtained metrics reduced to kno wn or flat ones. The developed mec hanism for the constructing in tegrable metrics ma y be further extended by considering pro jective equations (2.3), including its autonomous case (2.5), whic h can b e linearised b y non-autonomous transforma- tions (3.2). Moreo ver, we hav e introduced the notion of the generalized Darb oux in tegrabilit y in the context of b oth pro jectiv e v ector fields and geodesic flo ws and demonstrate that b oth families of metrics obtained in Section 5 are in tegrable in this sense. W e b eliev e that the concept of the generalized Darb oux in tegrability can be useful for constructing and classifying in tegrable geo desic flo ws with the help of the relative Killing v ectors and/or the inv ariants of the pro jective v ector fields. 23 7 Ac kno wledgmen ts J.G. is partially supp orted b y the Agencia Estatal de Inv estigac ´ ıon grant PID2020-113758GB- I00 and A GAUR grant num b er 2021SGR 01618. D.S. is partially supp orted b y H2020-MSCA- COFUND-2020-101034228-W OLFRAM2. Authors are grateful to anon ymous referees for their v aluable comments and suggestions. References [1] R.L. Bry ant, G. Manno, V.S. Matv eev, A solution of a problem of Soph us Lie: normal forms of t wo-dimensional metrics admitting tw o pro jective v ector fields, Math. Ann. 340 (2008) 437–463. doi:10.1007/s00208-007-0158-3. [2] V.S. Matveev, V. V. Shevc hishin, Tw o-dimensional sup erintegrable metrics with one linear and one cubic integral, J. Geom. Phys. 61 (2011) 1353–1377. doi:10.1016/j.geomphys.2011.02.012. [3] A. Bolsinov, V.S. Matv eev, E. Miranda, S. T abac hniko v, Open problems, questions and c hal- lenges in finite-dimensional in tegrable systems, Philos. T rans. R. So c. A Math. Phys. Eng. Sci. 376 (2018). doi:10.1098/rsta.2017.0430. [4] K. Burns, V.S. Matveev, Op en problems and questions ab out geo desics, Ergo d. Theory Dyn. Syst. (2019) 641–684. doi:10.1017/etds.2019.73. [5] V. V. Kozlo v, On rational integrals of geo desic flows, Regul. Chaotic Dyn. 19 (2014) 601–606. doi:10.1134/S156035471406001X. [6] G. V alent, C. Duv al, V. Shev chishin, Explicit metrics for a class of tw o- dimensional cubically superintegrable systems, J. Geom. Ph ys. 87 (2015) 461–481. doi:10.1016/j.geomph ys.2014.08.004. [7] B. Kruglik o v, In v ariant characterization of Liouville metrics and p olynomial integrals, J. Geom. Ph ys. 58 (2008) 979–995. doi:10.1016/j.geomphys.2008.03.005. [8] B. Kruglik ov, V.S. Matveev, The geodesic flow of a generic metric do es not admit non triv- ial in tegrals p olynomial in momenta, Nonlinearity . 29 (2016) 1755–1768. doi:10.1088/0951- 7715/29/6/1755. [9] G. V alent, Sup erintegrable mo dels on Riemannian surfaces of revolution with integrals of an y in teger degree (I), Regul. Chaotic Dyn. 22 (2017) 319–352. doi:10.1134/S1560354717040013. [10] Y.Y. Bagderina, Rational integrals of the second degree of t wo-dimen tional geo desic equations, Sib. Electron. Math. Rep orts. 14 (2017) 33–40. doi:10.17377/semi.2017.14.005. [11] S. Agap o v, V. Shubin, Rational in tegrals of 2-dimensional geo desic flows: New examples, J. Geom. Ph ys. 170 (2021) 104389. doi:10.1016/j.geomphys.2021.104389. [12] B. Kruglik ov, Rational First In tegrals and Relative Killing T ensors, arXiv:2412.04151 (2024) 1–11. doi:10.48550/arXiv.2412.04151 [13] B. Krugliko v, On F ractional-Linear In tegrals of Geo desics on Surfaces, Russ. J. Math. Phys. 32 (2025) 97–104. doi:10.1134/S1061920824601836. 24 [14] B. Kruglik ov, Rational first in tegrals and relative Killing tensors, J. Geom. Ph ys. 225 (2026) 105819. h ttps://doi.org/10.1016/j.geomphys.2026.105819. [15] J. Gin´ e, D.I. Sinelshc hiko v, On the geometric and analytical prop erties of the an- harmonic oscillator, Commun. Nonlinear Sci. Numer. Simul. 131 (2024) 107875. doi: 10.1016/j.cnsns.2024.107875 [16] S. V Agap ov, M. V Demina, In tegrable geo desic flows and metrisable second- order ordinary differen tial equations, J. Geom. Phys. 199 (2024) 105168. doi: 10.1016/j.geomph ys.2024.105168. [17] S. V. Agap ov, A.E. Mirono v, Finite-Gap P otentials and Integrable Geo desic Equations on a 2-Surface, Pro c. Steklo v Inst. Math. 327 (2024) 1–11. doi:10.1134/S0081543824060014. [18] D.I. Sinelshc hiko v, Linearizability conditions for the Rayleigh-lik e oscillators, Phys. Lett. A. 384 (2020) 126655. doi:10.1016/j.ph ysleta.2020.126655. [19] D.I. Sinelshc hik ov, On linearizabilit y via nonlocal transformations and first integrals for second-order ordinary differen tial equations, Chaos Soliton. F ract. 141 (2020) 110318. doi:10.1016/j.c haos.2020.110318. [20] F. Dumortier, J. Llibre, J.C. Art ´ es, Qualitative Theory of Planar Differential Systems, Springer Berlin Heidelb erg, 2006. doi:10.1007/978-3-540-32902-2. [21] X. Zhang, In tegrabilit y of Dynamical Systems: Algebra and Analysis, Springer, Singap ore, 2017. doi:10.1007/978-981-10-4226-3. [22] A. Goriely , Integrabilit y and nonin tegrabilit y of dynamical systems, W orld Scientific, 2001. [23] I.A. Garc ´ ıa, J. Gin´ e, Generalized cofactors and nonlinear sup erp osition principles, Appl. Math. Lett. 16 (2003) 1137–1141. h ttps://doi.org/10.1016/S0893-9659(03)90107-8. [24] J. Gin´ e, M. Grau, W eierstrass in tegrability of differential equations, Appl. Math. Lett. 23 (2010) 523–526. h ttps://doi.org/10.1016/j.aml.2010.01.004. [25] J. Gin ´ e, D. Sinelshc hiko v, Integrabilit y of Oscillators and T ranscenden tal Inv ariant Curves, Qual. Theory Dyn. Syst. 24 (2025) 26. doi:10.1007/s12346-024-01182-x. [26] B.A. Dubro vin, A.T. F omenko, S.P . No vik ov, Mo dern Geometry — Metho ds and Applications. P art I. The Geometry of Surfaces, T ransformation Groups, and Fields, Springer New Y ork, New Y ork, NY, 1984. https://doi.org/10.1007/978-1-4684-9946-9. [27] R. Bry ant, M. Duna jski, M. Eastw o o d, Metrisability of t wo-dimensional pro jective structures, J. Differ. Geom. 83 (2009) 465–500. doi:10.4310/jdg/1264601033. [28] R. Liouville, Sur les in v arian ts de certaines ´ equations diff´ eren tielles et sur leurs applications, J. l’ ´ Ecole P olytech. 59 (1889) 7–76. [29] F. Contatto, M. Duna jski, Metrisability of Painlev ´ e equations, J. Math. Ph ys. 59 (2018) 023507. doi:10.1063/1.4998147. [30] V.S. Matv eev, Two-dimensional metrics admitting precisely one pro jectiv e v ector field, Math. Ann. 352 (2012) 865–909. h ttps://doi.org/10.1007/s00208-011-0659-y . 25 [31] A. T resse, D´ etermination des in v arian ts p onctuels de l’ ´ equation diff´ eren tielle ordinaire du second ordre y , S. Hirzel, 1896. [32] B. Krugliko v, Poin t Classification of Second Order ODEs: T resse Classification Revisited and Bey ond, in: Differ. Equations - Geom. Symmetries In tegr., Springer Berlin Heidelberg, Berlin, Heidelb erg, 2009: pp. 199–221. doi:10.1007/978-3-642-00873-3 10. [33] V.A. Y umaguzhin, Differen tial in v arian ts of second order ODEs, I, Acta Appl. Math. 109 (2010) 283–313. doi:10.1007/s10440-009-9454-0. [34] Y.Y. Bagderina, Inv arian ts of a family of scalar second-order ordinary differen tial equa- tions for Lie symmetries and first in tegrals, J. Ph ys. A Math. Theor. 49 (2016) 155202. doi:10.1088/1751-8113/49/15/155202. [35] A.J. Maciejewski, M. Przybylsk a, Darb oux p olynomials and first in tegrals of natural p olyno- mial Hamiltonian systems, Ph ys. Lett. Sect. A Gen. A t. Solid State Ph ys. 326 (2004) 219–226. h ttps://doi.org/10.1016/j.physleta.2004.04.034. [36] A. Aoki, T. Houri, K. T omo da, Rational first in tegrals of geo desic equations and generalised hidden symmetries, Class. Quan tum Gra vity 33 (2016) 195003. h ttps://doi.org/10.1088/0264- 9381/33/19/195003. [37] V.I. Arnold, Mathematical Methods of Classical Mechanics, second edition, Springer New Y ork, New Y ork, NY, 1989. https://doi.org/10.1007/978-1-4757-2063-1. [38] J.F. Cari˜ nena, E. Mart ´ ınez, M.C. Mu˜ noz-Lecanda, Infinitesimal Time Reparametri- sation and Its Applications, J. Nonlinear Math. Phys. 29 (2022) 523–555. h ttps://doi.org/10.1007/s44198-022-00037-w. [39] D. Sinelshc hiko v, On an in tegrabilit y criterion for a family of cubic oscillators, AIMS Math. 6 (2021) 12902–12910. h ttps://doi.org/10.3934/math.2021745. [40] G. Ko enigs, Sur les g´ eo desiques a int ´ egrales quadratiques. Note I I from Darb oux’ ‘Le¸ cons sur la th ´ eorie g´ en ´ erale des surfaces’, v ol. IV. Chelsea Publishing (1896); Bronx, N.Y., 1972 [41] J. Gin ´ e, J. Llibre, Integrabilit y and algebraic limit cycles for p olynomial differentia l sys- tems with homogeneous nonlinearities, J. Differ. Equ. 197 (2004) 147–161. doi:10.1016/S0022- 0396(03)00199-2. [42] E. Kamk e, Differentialgleic h ungen L¨ osungsmetho den und L¨ osungen, View eg+T eubner V erlag, Wiesbaden, 1977. doi:10.1007/978-3-663-05925-7. [43] A. V Bolsino v, V.S. Matveev, A.T. F omenko, Tw o-dimensional Riemannian metrics with in tegrable geo desic flo ws. Lo cal and global geometry , Sb. Math. 189 (1998) 1441–1466. doi:10.1070/sm1998v189n10ab eh000346. 26