The transformations between N=2 supersymmetric KdV and HD equations

The transfo rmations b et w een N = 2 sup ersymmetr ic KdV and HD equations Kai Tian and Q. P . Liu 1 Dep artment of Mathematics, China University of Mining and T e chnolo gy, Beijing 100083, P. R. China (Dated: 29 Ma y 2022) The N = 2 sup ercomformal tra nsformations ar e emplo y ed to study supersymmetric in tegrable sy stems. It is pro v ed that t w o known N = 2 supersymmetric Harry Dym equations are transformed in to tw o N = 2 supersymmetric mo dified Kort w eg-de V ries equations, th us ar e connected with t w o N = 2 sup ersymmetric Kort w eg-de V ries eq uations. 1 I. INTRODUCTION The recipro cal transformation, also known as ho dograph transfor ma t io n, plays an im- p ortant role when w e in ve stigate relations a mong some no nlinear ev olution equations. F or instance, the Harry Dym (HD) equation (or hierar ch y) 4 , whic h is in v ariant under a kind o f re- cipro cal transfor ma t io n, is also recipro cally linked to the Kortew eg-de V ries (KdV) equation (or hierarc h y). The Camassa-Holm equation is show n t o b e link ed to the first negativ e flo w of the KdV hierarc hy 6 . The Kaw amoto equation is transformed to the common mo dification of the Sa wada-Kotera and Kaup-Kup ershmidt equations b y recip ro cal transformation 7 and man y other examples exist. Apart from its o wn in terest, t he recipro cal tr a nsformation is also a p o w erful to o l to inv estigate in tegrable prop erties of nonlinear ev olution equations. Indeed, recursion op erators, bi-Hamiltonian structures and solutions of one giv en equation could b e constructed from the corresponding properties of the other equation through the asso ciated recipro cal transformat io n. With the help of the r ecipro cal transformation, the abundan t symmetry structures of Kaw a mo t o -t yp e equations a nd Harry Dym equation w ere rev ealed 5 . V ery recen tly , the recipro cal transformation w as generalized to N = 1 sup ersymmetric equations 12 , where a general pro cedure to construct supersymmetric rec ipro cal transforma- tion w as presen ted. As a pplications, one of the sup ersymmetric HD eq uations w as sho wn to b e recipro cally link ed to the sup ersymmetric mo dified KdV equation, and the super- symmetric Kaw amot o equation, as a fifth order analog of HD equation, w as transformed to the sup ersymmetric mo dified Sa w ada- Kotera equation. As in the classic al case , the su- p ersymmetric recipro cal transformation could b e employ ed to explore inte gra ble prop erties of supersymmetric eq uatio ns, whic h w as illus trat ed b y constructing the recursion op erato r s and bi-Hamiltonian structure s o f the sup ersymmetric HD and Ka w amoto equations. More examples of sup ersymmetric rec ipro cal transformation can b e found in Ref. 13. Besides N = 1 supersymmetric g eneralizatio ns, the in tegrable s ystems also admit N = 2 extended sup ersymmetric generalizations. The idea could almost b e traced back to t he usage of the sup ersymmetry in t he quan tum field theory 8 . As a striking feature, N = 2 extended case distinguishes itself fro m N = 1 non- extended case b y the p ossibilit y to supply new classical in tegrable systems. The N = 2 sup ersymmetric KdV equations w ere prop osed more tha n t we nt y ye ars ago 9,10 and hav e b een studied extensiv ely since then. V arious r esults 2 for these equations ha ve b een o bta ined, including Lax represen ta t io ns 10 ,16 , bi- Hamiltonian structures 15 , bilinear form ulism 18 and so on. In addition to the KdV equation, some other in tegrable equations, suc h as HD equation, Camassa-Holm equation w ere also generalized to the N = 2 sup er space 3,17 (see also Refs. 1 and 11). With the success of establishing the N = 1 sup ersymmetric recipro cal transformation, it w ould b e imp ortant to extend the results to the N = 2 sp ersymme tric case and to study the p ossible links b et w een N = 2 sup ersymmetric HD and K dV eq uatio ns. In this pap er, w e will sho w that suc h extension is indeed p ossible. W e will sho w that the sup ercomfor ma l transformations serv e our need. The pap er is organized as follo ws. In the next se ction, w e recall the N = 2 supersymmetric KdV and HD equations. In the section three, the N = 2 sup ercomformal transformations and the rele v an t results will b e review ed and elaborat ed. Then in the section four, the transformatio ns b etw een t he tw o N = 2 sup ersymmetric HD eq uations and sup ersymme tric KdV eq uations are giv en. Final sec tion con tains some commen ts and o p en questions. I I. N = 2 SUPERSYMMETRIC KDV AND HD EQUA TION S The N = 2 sup ersymmetric KdV equation is g iv en by the one-parameter system Φ τ = 1 4  Φ 3 y − 3[Φ( D 1 D 2 Φ)] y − a − 1 2 [ D 1 D 2 Φ 2 ] y − 3 a Φ 2 Φ y  , (1) whic h is denoted b y SKdV a equation in literature. Some con ve ntional criteria ha v e b een adopted to study its integrabilit y and the system is kno wn to b e in tegrable only for certain v alues of the parameter a . In fact, the existence of infinitely many conserv ation law s implies that a only takes three v alues, − 2, 1 and 4 . 10 The singularity analysis also leads to the same conclusion 2 . F or these three cases, Eq. (1) w as show n to admit Lax represen tations 10 ,16 a = 4 : L 4 = − ( D 1 D 2 + Φ) 2 , ∂ ∂ τ n L 4 = [( ˆ L n 4 L 1 2 4 ) ≥ 0 , L 4 ] , a = − 2 : L − 2 = ∂ 2 y + D 1 Φ D 2 − D 2 Φ D 1 , ∂ ∂ τ n L − 2 = [( L n 2 − 2 ) ≥ 0 , L − 2 ] , a = 1 : L 1 = − ∂ − 1 y D 1 D 2 ( D 1 D 2 + Φ) , ∂ ∂ τ n L 1 = [( L 3 n 1 ) ≥ 1 , L 1 ] , where we are us ing the standar d nota t ions and the subscripts ≥ k denote the corresp o nding pro jections. It is remark ed that that the operator L 4 p ossesses tw o differen t square ro ots, 3 namely ˆ L 4 = i ( D 1 D 2 + Φ) , and L 1 2 4 = ∂ y + · · · . The non-uniqueness of ro ots of Lax op erator results in the SKdV 4 equation whic h admits t wice as man y conserv ed quan tities as those of the other tw o systems 15 . It is easy to see, from the Lax represen tation, t ha t the SKdV 4 equation is not the first non- trivial flow in its hierarc h y , while the first one is Φ τ 1 = 1 2 ( D 1 D 2 Φ y ) + 2ΦΦ y . (2) Let us no w turn to the Harry Dym case. By considering the most general N = 2 super Lax op erator whic h consists of differential op erators only , t w o differen t N = 2 sup ersymmetric Harry Dym hierarchie s w ere presen ted 3 . One is formulated a s ∂ ∂ t n L 1 = h  ˆ L n 1 L 1 2 1  ≥ 2 , L 1 i , n = 0 , 1 , 2 . · · · (3) where the Lax op erator is L 1 = − ( W D 1 D 2 ) 2 , (4) whose t wo diff erent square ro ots are g iven b y ˆ L 1 = iW D 1 D 2 , L 1 2 1 = W ∂ x + 1 2 h ( D 1 W ) D 1 + ( D 2 W ) D 2 − W x i + 1 4 h − 2( D 1 D 2 W ) − ( D 2 W )( D 1 W ) W − 1 i D 1 D 2 ∂ − 1 x + · · · · · · , resp ectiv ely . Based on the L a x equation (3), it is easy to write dow n the first tw o non-trivial flo ws in this hierarch y explicitly , namely , W t 1 = i 2 ( D 1 D 2 W x ) W 2 , (5) W t 2 = 1 8  2 W 3 x W 3 − 6( D 1 D 2 W x )( D 1 D 2 W ) W 2 − 3( D 2 W 2 x )( D 2 W ) W 2 − 3( D 1 W 2 x )( D 1 W ) W 2  . (6) The other N = 2 sup ersymmetric Harry Dym hierarc h y is defined as ∂ ∂ t n L 2 = h  L n 2 2  ≥ 2 , L 2 i (7) with the Lax op erator L 2 = 1 2 ( D 1 W 2 D 1 + D 2 W 2 D 2 ) ∂ x . (8) 4 The first non-tr ivial flow o f this hierarch y reads as W t 3 = 1 8  2 W 3 x W 3 − 3( D 2 W 2 x )( D 2 W ) W 2 − 3( D 1 W 2 x )( D 1 W ) W 2 + 3( D 2 W )( D 1 W )( D 1 D 2 W x ) W  . (9) In b oth cases, W = W ( x, θ 1 , θ 2 , t ) is a b osonic sup er field. I I I. THE SUPERCONF OR MAL TRANSFORMA TION OF N = 2 SUPER SP A CE A super diffeomorphism b et w een t wo sup er space s is named as a superconforma l trans- formation providing the sup er deriv ative s a re transformed co v a r ian tly (see Ref. 14 and the references there). Let us consider the sup er diffeomorphism b etw een the s up er spaces ( y ,  1 ,  2 ) and ( x, θ 1 , θ 2 ) y → x = x ( y ,  1 ,  2 ) ,  1 → θ 1 = θ 1 ( y ,  1 ,  2 ) ,  2 → θ 2 = θ 2 ( y ,  1 ,  2 ) . (10) The asso ciated sup er de riv ativ es for eac h sup er space are resp ectiv ely denoted b y D k = ∂  k +  k ∂ y , D k = ∂ θ k + θ k ∂ x , ( k = 1 , 2 ) . F or the sup er diffeomorphism (10), w e ha ve the follo wing relatio ns D 1 =  ( D 1 x ) − θ 1 ( D 1 θ 1 ) − θ 2 ( D 1 θ 2 )  ∂ ∂ x + ( D 1 θ 1 ) D 1 + ( D 1 θ 2 ) D 2 , (11a) D 2 =  ( D 2 x ) − θ 1 ( D 2 θ 1 ) − θ 2 ( D 2 θ 2 )  ∂ ∂ x + ( D 2 θ 1 ) D 1 + ( D 2 θ 2 ) D 2 , (11b) hence, to ensure the sup er deriv ativ es tra nsforming co v arian tly , i.e. D 1 = ( D 1 θ 1 ) D 1 + ( D 1 θ 2 ) D 2 , D 2 = ( D 2 θ 1 ) D 1 + ( D 2 θ 2 ) D 2 , (12) the sup er diffeomorphism ( 1 0) m ust b e constrained by ( D 1 x ) = θ 1 ( D 1 θ 1 ) + θ 2 ( D 1 θ 2 ) , ( D 2 x ) = θ 1 ( D 2 θ 1 ) + θ 2 ( D 2 θ 2 ) . (13) Ho w ev er, the constrain ts (13) are not sufficien t to ensure the sup er diffeomorphism to b e sup erconformal. Some further constraints are resulted fro m the relation D 2 1 = D 2 2 = ∂ x . 5 Through cum b ersome, otherwise straigh tforward calculation, one obtains ( D 1 θ 2 ) = − ( D 2 θ 1 ) , ( D 2 θ 2 ) = ( D 1 θ 1 ) . (14) Summing up, w e emphasize that under the constraints (13) and (14), the super diffeo- morphism (10) is a sup erconfor mal transformation, whic h can b e form ulated b y D 1 = K − 1  ( D 1 θ 1 ) D 1 + ( D 2 θ 1 ) D 2  , (15a) D 2 = K − 1  − ( D 2 θ 1 ) D 1 + ( D 1 θ 1 ) D 2  , (15b) where K = ( D 1 θ 1 ) 2 + ( D 2 θ 1 ) 2 . The follo wing fo rm ulas, whic h can be c heck ed directly , D 1 D 2 = K − 1  D 1 D 2 + 1 2 ( D 2 log K ) D 1 − 1 2 ( D 1 log K ) D 2  , (16) ∂ x = K − 1  ∂ y − 1 2 ( D 2 log K ) D 2 − 1 2 ( D 1 log K ) D 1  , (17) are very useful when w e study the link betw een the N = 2 supersymmetric HD hierarc hies to sup ersymmetric MKdV t ype hie rar chies . IV. RECIP R OCAL TRANSF ORMA TI ONS In this section, we show that b oth N = 2 sup ersymmetric HD hierarchie s mentioned in the section t w o are link ed with sup ersymmetric MKdV t yp e hierarc hies via sup erconformal transformations. First, let us consider the b eha vior of the Lax op erators L 1 and L 2 under certain sup erconformal transformations. F or the Lax o p erator L 1 = − ( W D 1 D 2 ) 2 , if w e assume W = K , then according to the form ula ( 1 6), w e hav e the new op era t or in the sup er space ( y ,  1 ,  2 ) L 1 m = −  D 1 D 2 + 1 2 ( D 2 log K ) D 1 − 1 2 ( D 1 log K ) D 2  2 . (18) Let U = 1 / 2 log K , then L 1 m = − [ D 1 D 2 + ( D 2 U ) D 1 − ( D 1 U ) D 2 ] 2 . (19) Applying the ga ug e transformation on L 1 m , w e hav e e − U L 1 m e U = − [ D 1 D 2 + ( D 1 D 2 U ) + ( D 2 U )( D 1 U )] 2 ≡ − [ D 1 D 2 + Φ] 2 6 whic h is nothing but the Lax op erator L 4 of SKdV 4 equation. This implies that L 1 m should b e a Lax op erator for the mo dification of SKdV 4 equation. In fact, w e hav e Prop osition 1 The L ax e quation ∂ ∂ τ n L 1 m = h  ˆ L n 1 m L 1 2 1 m  ≥ 1 , L 1 m i , n = 0 , 1 , 2 , · · · (20) defines a N = 2 sup ersymmetric hie r ar chy, wher e ˆ L 1 m = i [ D 1 D 2 + ( D 2 U ) D 1 − ( D 1 U ) D 2 ] , and L 1 / 2 1 m = ∂ y + · · · · · · . The first two non-trivial flows in this hie r ar chy ar e explicitly given by U τ 1 = i 2 h ( D 1 D 2 U y ) + 2( D 1 D 2 U ) U y + 2( D 2 U y )( D 1 U ) + 2 ( D 2 U )( D 1 U ) i , (21) U τ 2 = 1 4 h U 3 y − 2 U 3 y − 6( D 1 D 2 U y )( D 1 D 2 U ) − 6( D 1 D 2 U ) 2 U y − 3( D 2 U y )( D 2 U ) U y − 9( D 2 U y )( D 1 U )( D 1 D 2 U ) − 9( D 2 U )( D 1 U y )( D 1 D 2 U ) − 6( D 2 U )( D 1 U )( D 1 D 2 U y ) − 3( D 1 U y )( D 1 U ) U y i . (22) F urthermor e, the τ 2 flow (22) is the mo dific ation of SKdV 4 e quation with the Miur a tr a n s- formation Φ = ( D 1 D 2 U ) + ( D 2 U )( D 1 U ) . Pr o of : Direct calculations. The sup erconformal transformation provides us a link b et w een the spatial v ariables. T o ha v e a complete picture, we need to find the counterpart fo r the temp oral v ariables. F or the couple of Lax hierarc hies (3) and (20), the relations a re giv en by ∂ ∂ t n −  ˆ L n 1 L 1 2 1  ≥ 2 = ∂ ∂ τ n −  ˆ L n 1 m L 1 2 1 m  ≥ 1 . (23) W e no w calculate the explicit tra nsformations fo r t he first and second flows . In the simplest case, namely n = 1, since  ˆ L 1 L 1 2 1  ≥ 2 = iW 2 ∂ x D 1 D 2 + i 2 ( D 1 W ) W ∂ x D 2 − i 2 ( D 2 W ) W ∂ x D 1 + i 2 W D 1 D 2 = i∂ y D 1 D 2 − i ( D 1 U ) ∂ y D 2 + i ( D 2 U ) ∂ y D 1 − iU y D 1 D 2 +  − i ( D 1 U y ) + i ( D 1 U ) U y  D 2 +  i ( D 2 U y ) − i ( D 2 U ) U y  D 1 ,  ˆ L 1 m L 1 / 2 1 m  ≥ 1 = i∂ y D 1 D 2 − i ( D 1 U ) ∂ y D 2 + i ( D 2 U ) D 1 + i  ( D 1 D 2 U ) + ( D 2 U )( D 1 U )  ∂ y − iU y D 1 D 2 7 + 1 2  − 3 i ( D 1 U y ) + 3 i ( D 1 U ) U y + i ( D 2 U )( D 1 D 2 U )  D 2 + 1 2  3 i ( D 2 U y ) − 3 i ( D 2 U ) U y + i ( D 1 U )( D 1 D 2 U )  D 1 . Hence, w e hav e ∂ ∂ t 1 = ∂ ∂ τ 1 − i  ( D 1 D 2 U ) + ( D 2 U )( D 1 U )  ∂ y + i 2  ( D 1 U y ) − ( D 1 U ) U y − ( D 2 U )( D 1 D 2 U )  D 2 + i 2  − ( D 2 U y ) + ( D 2 U ) U y − ( D 1 U )( D 1 D 2 U )  D 1 . Similarly , when n = 2, we hav e ∂ ∂ t 2 = ∂ ∂ τ 2 +  ˆ L 2 1 L 1 2 1  ≥ 2 −  ˆ L 2 1 m L 1 2 1 m  ≥ 1 = ∂ ∂ τ 2 + 1 2  3( D 1 D 2 U ) 2 − U 2 y + U 2 y + ( D 2 U y )( D 2 U ) + 4 ( D 2 U )( D 1 U )( D 1 D 2 U ) + ( D 1 U y )( D 1 U )  ∂ y + 1 4  − ( D 2 U 2 y ) + ( D 2 U y ) U y + 4( D 2 U )( D 1 D 2 U ) 2 + ( D 2 U ) U 2 y + 4( D 2 U )( D 1 U y )( D 1 U ) − 5( D 1 U y )( D 1 D 2 U ) − ( D 1 U )( D 1 D 2 U y ) + 4( D 1 U )( D 1 D 2 U ) U y  D 2 + 1 4  − ( D 1 U 2 y ) + ( D 1 U y ) U y + 4( D 1 U )( D 1 D 2 U ) 2 + ( D 1 U ) U 2 y + 4( D 2 U y )( D 2 U )( D 1 U ) + 5( D 2 U y )( D 1 D 2 U ) + ( D 2 U )( D 1 D 2 U y ) − 4( D 2 U )( D 1 D 2 U ) U y  D 1 . Let us no w consider the other N = 2 sup ersymmetric Harry Dym eq uatio n, whose Lax op erator is L 2 = 1 / 2( D 1 W 2 D 1 + D 2 W 2 D 2 ) ∂ x . In this case, w e also assume W = K . F rom the form ulas ( 1 5a) (15b), w e ha v e 1 2 ( D 1 W 2 D 1 + D 2 W 2 D 2 ) = K ∂ y + 1 2 ( D 2 K ) D 2 + 1 2 ( D 1 K ) D 1 . Then taking the formula (17) in to account, L 2 is transformed to L 2 m =  K ∂ y + 1 2 ( D 2 K ) D 2 + 1 2 ( D 1 K ) D 1  K − 1  ∂ y − 1 2 ( D 2 log K ) D 2 − 1 2 ( D 1 log K ) D 1  = ∂ 2 y − U y ∂ y − 1 2 ( D 2 U )( D 1 U ) D 1 D 2 + 1 4  − 2( D 2 U y ) + ( D 2 U ) U y − ( D 1 U )( D 1 D 2 U )  D 2 + 1 4  − 2( D 1 U y ) + ( D 1 U ) U y + ( D 2 U )( D 1 D 2 U )  D 1 . (24) By direct calculation, w e hav e 8 Prop osition 2 The L ax e quation ∂ ∂ τ n L 2 m = h  L n 2 2 m  ≥ 1 , L 2 m i , n = 1 , 3 , · · · (25) defines a N = 2 sup ersymmetric hie r ar chy, whose first non-trivial flow is U τ 3 = 1 16  4 U 3 y − 2 U 3 y − 3( D 2 U y )( D 2 U ) U y + 3( D 2 U y )( D 1 U )( D 1 D 2 U ) + 3( D 2 U )( D 1 U y )( D 1 D 2 U ) − 3( D 1 U y )( D 1 U ) U y  . (26) F urthermor e, the e quation (26) is a mo dific ation of the SKdV − 2 e quation with the Miur a typ e tr ansform a tion Φ = 1 2 ( D 1 D 2 U ) + 1 4 ( D 2 U )( D 1 U ) . (27) Pr o of : Direct calculations. Remark: As a b ypro duct of ab ov e results, a relat io n could b e inferred b et w een the b osonic limit of SKdV − 2 and tha t of Eq. (26). Let Φ = φ 0 + θ 2 θ 1 φ 1 and U = u 0 + θ 2 θ 1 u 1 , then the b osonic limits of SKdV − 2 and Eq. (26) a r e resp ectiv ely giv en b y φ 0 ,τ 3 = 1 4  φ 0 , 3 y + 6 φ 0 ,y φ 2 0  , φ 1 ,τ 3 = 1 4  φ 1 , 3 y − 6 φ 1 ,y φ 1 + 6 φ 1 ,y φ 2 0 + 12 φ 1 φ 0 ,y φ 0 − 6 φ 0 , 2 y φ 0 ,y  , and u 0 ,τ 3 = 1 8  2 u 0 , 3 y − u 3 0 ,y  , u 1 ,τ 3 = 1 8  2 u 1 , 3 y + 3 u 1 ,y u 2 1  . The Miura transformation betw een t hem is φ 0 = 1 2 u 1 , φ 1 = 1 4  − 2 u 0 , 2 y + u 2 0 ,y + u 2 1  . As abov e, w e deriv e the transformation b et w een the temp oral v ariables. F or the couple of hierarch ies (7) a nd (25), t he relations b et w een v ector fields of t ime v ariables a re giv en b y ∂ ∂ t n −  L n 2 2  ≥ 2 = ∂ ∂ τ n −  L n 2 2 m  ≥ 1 . (28) When n = 3, we ha v e ∂ ∂ t 3 = ∂ ∂ τ 3 +  L 3 2 2  ≥ 2 −  L 3 2 2 m  ≥ 1 = ∂ ∂ τ 3 + 1 8  − 2 U 2 y + U 2 y + ( D 2 U y )( D 2 U ) − ( D 2 U )( D 1 U )( D 1 D 2 U ) + ( D 1 U y )( D 1 U )  ∂ y 9 + 1 16  − 2( D 2 U 2 y ) + ( D 2 U y ) U y − ( D 2 U )( D 1 D 2 U ) 2 + ( D 2 U ) U 2 y − ( D 2 U )( D 1 U y )( D 1 U ) + ( D 1 U y )( D 1 D 2 U ) − ( D 1 U )( D 1 D 2 U y ) − ( D 1 U )( D 1 D 2 U ) U y  D 2 + 1 16  − 2( D 1 U 2 y ) + ( D 1 U y ) U y − ( D 1 U )( D 1 D 2 U ) 2 + ( D 1 U ) U 2 y − ( D 2 U y )( D 2 U )( D 1 U ) − ( D 2 U y )( D 1 D 2 U ) + ( D 2 U )( D 1 D 2 U y ) + ( D 2 U )( D 1 D 2 U ) U y  D 1 . V. SUMMAR Y AND P R OBLEMS With the help of supercomformal transformations, w e ha v e es tablished the relations b e- t w een t wo N = 2 supersymmetric K dV equations and t w o sup ersymmetric HD equations, therefore w e generalize our results of N = 1 supersymmetric rec ipro cal transformations to the N = 2 case. While this is in teresting, there are a num b er of problems to b e solv ed. W e list some of them as follo ws • Our construction ab o v e relies on the Lax represen tations, so it w ould b e imp orta n t to reco v er the tr a nsformations b y means of o ther prop erty suc h as conserv ation la ws. • Studying the implications of our transformations is anot her in teresting problem. In- deed, mo r e prop erties are kno wn for the KdV cases than for the HD cases, for instance, bi-Hamiltonian structures hav e been constructed for the N = 2 sup ersymmetric KdV systems , but w e kno w little a b out Hamiltonian s tructures for the N = 2 supersym- metric HD equations apa r t f orm the following observ ation. The system s (5) and (6) can b e reformulated as W t 1 = B δ H 1 δ W , W t 2 = B δ H 2 δ W , where B = W 2 D 1 D 2 ∂ x W 2 is a Hamiltonian op erator and Hamiltonian functionals are giv en by H 1 = i 2 Z ln W d z d θ 1 d θ 2 , and H 2 = 1 8 Z ( D 1 W )( D 2 W ) W − 1 d z d θ 1 d θ 2 . • In the Harry Dym case, t w o kno wn N = 2 in tegrable sup ersymmetric extensions are pro v ed to b e link ed with tw o N = 2 sup ersymmetric KdV equations, but there are three rather than t w o suc h systems. Thus , this fact seems to indicate that o ne N = 2 in tegrable supersymmetric Harr y Dym equation is missing. 10 The progress on solving thes e and other related pr o blems ma y b e rep orted elsewhere. Ac knowle dgmen t This w ork is suppor t ed b y the National Natural Science F oundation of China (gran t n um b ers: 10731080, 1097 1 222) and the F undamen tal Researc h F unds for Cen tral Univ ersities. REFEREN CES 1 H. Aratyn, J. .F. Gomes a nd A. 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