Integrable pseudopotentials related to elliptic curves

In tegrable pseudop oten tials related to elliptic curv es A.V. Odesskii 1 , 2 , V.V. Sok olov 2 1 Bro ck Univ ersity (Canada) 2 Landau Institute for Theoretica l Physics (Russia) Abstract W e construct in tegrable pseudop oten tials with an arbitrary n um b er of fields in terms of e lliptic generalization of h yp ergeometric functions in sev eral v ariables. These pseudop o ten tials yield some in tegrable (2+1)-dimensional h ydro dynamic t yp e systems. An in teresting c lass of in tegrable (1+1 ) - dimensional h ydro dynamic type systems is also generated by our pseudopo - ten tials. MSC n umbers: 17B80 , 17B63, 32L81, 14H70 Address : L.D. Landau Institute for Theoretical Phy sics of Russian Academ y of Sciences, Kosygina 2, 1 1 9334, Mosco w, Russia E-mail : ao dessk i@bro c ku.ca, sok olov@itp.ac.ru 1 Con t e n ts 1 In t r oduct ion 3 2 Elliptic hypergeometric functions 6 3 Elliptic pseudop oten tials of defect 0 7 4 Elliptic pseudop oten tials of defect k > 0 10 5 In t egr able (1+1)-dimensional h ydro dynamic-t yp e systems and h ydro dynamic reductions 14 2 1 In tro duction In [1] a wide class of 3-dimensional integrable PDEs o f the form m X j =1 a ij ( u ) u j,t 1 + m X j =1 b ij ( u ) u j,t 2 + m X j =1 c ij ( u ) u j,t 3 = 0 , i = 1 , ..., l, (1.1) where u = ( u 1 , . . . , u m ) was constructed. The co efficien t s of these PDEs w ere written in terms of generalized hypergeometric functions [2]. By the in tegrability of (1.1) w e mean the existence of a pseudop oten tia l represen tation 1 ψ t 2 = A ( p, u ) , ψ t 3 = B ( p, u ) , where p = ψ t 1 . (1.2) Suc h a pseudop oten t ia l represen tat io n is a dis p ersionless v ersion [3, 4] of the zero curv ature represen tation, whic h is a basic notion in the integrabilit y theory of solitonic equations (see [5]). One of the in teresting and attractiv e features of the theory of integrable systems (1.1) is tha t the de p endence of the pseudop oten tials on p can b e m uc h more complicated then in the solitonic case. In [6, 7] some imp ortant examples of pseudop oten tials A, B related to the Whitham av eraging pro cedure for in tegrable disp ersion PD Es and to the F rob enious manifolds w ere fo und. These examples are related to t he univ ersal algebraic curve of g enus g with M punctures for arbitrary g , M . More precisely , the p oin t  A ppp A 2 pp , A p  runs o ver a curv e of genus g while p runs ov er C and u are some co or dinates o n the mo duli space M g ,M of curv es of genus g with M punctures . The pseudop oten tials fro m [1] (see also [8 ]) were written in the follo wing parametric form: A = F 1 ( ξ , u ) , p = F 2 ( ξ , u ) , where the ξ -dep endence of the functions F i is defined by the ODE F i,ξ = φ i ( ξ , u ) · ξ − s 1 ( ξ − 1) − s 2 ( ξ − u 1 ) − s 3 ... ( ξ − u m ) − s m +2 . (1.3) Here s 1 , ..., s m +2 are arbitr a ry constants and φ i are p olynomials in ξ of degree m − k . These pseudop o ten tials are relat ed to rational algebraic curv es. If s 1 = ... = s m +2 = 0 and k = 0 , then they coincide with pseudop oten tials from [6] related to M 0 ,m +3 . In this pap er w e construct integrable systems (1.1) and pseudopotentials related to elliptic curv e. F or these systems u = ( u 1 , . . . , u n , τ ), w here τ is the parameter of the e lliptic curv e. Note that τ is also an unkno wn function in our systems (1.1 ) . The co efficie n ts of the systems are e xpresse d in terms of some elliptic generalization of h yp ergeometric functions in sev eral v ariables. These elliptic h yp ergeometric functions can b e defined as solutio ns of the follo wing 1 This means that (1.1) is e quiv alen t to the compa tibilit y co nditions for (1.2). 3 compatible linear o v erdetermined system of PDEs: g u α u β = s β  ρ ( u β − u α ) + ρ ( u α + η ) − ρ ( u β ) − ρ ( η )  g u α + s α  ρ ( u α − u β ) + ρ ( u β + η ) − ρ ( u α ) − ρ ( η )  g u β , g u α u α = s α X β 6 = α  ρ ( u α ) + ρ ( η ) − ρ ( u α − u β ) − ρ ( u β + η )  g u β +  X β 6 = α s β ρ ( u α − u β ) + ( s α + 1) ρ ( u α + η ) + s α ρ ( − η ) + ( s 0 − s α − 1 ) ρ ( u α ) + 2 π ir  g u α − s 0 s α ( ρ ′ ( u α ) − ρ ′ ( η )) g , g τ = 1 2 π i X β  ρ ( u β + η ) − ρ ( η )  g u β − s 0 2 π i ρ ′ ( η ) g (1.4) for a single function g ( u 1 , . . . , u n , τ ) . Here and in the sequel η = s 1 u 1 + ... + s n u n + r τ + η 0 , s 0 = − s 1 − ... − s n , where s 1 , ..., s n , r, η 0 are arbitrary constan ts, and θ ( z ) = X α ∈ Z ( − 1) α e 2 π i ( αz + α ( α − 1) 2 τ ) , ρ ( z ) = θ ′ ( z ) θ ( z ) . (1.5) In the ab o v e formulas and in the sequel w e omit the second arg umen t τ of the functions θ , ρ and use the notation ρ ′ ( z ) = ∂ ρ ( z ) ∂ z , ρ τ ( z ) = ∂ ρ ( z ) ∂ τ , θ ′ ( z ) = ∂ θ ( z ) ∂ z , θ τ ( z ) = ∂ θ ( z ) ∂ τ . It turns o ut that the dimension of the space of solutions for (1.4) equals n + 1 . The pa p er is organized as follo ws. In Section 2 w e describ e some prop erties of elliptic h yp ergeometric functions needed for our purp oses. In particular, w e presen t a n integral represen t a tion similar to the represe n tatio n for the generalized h yp ergeometric function (see, for example [1]). In Section 3 for a n y n w e construct pseudop oten t ia ls (1.2) with k = 0 related t o the elliptic h yp ergeometric functions. The pseudop oten tial A n ( p, u 1 , ..., u n , τ ) is defined in a parametric form b y A n = P n ( g 1 , ξ ) , p = P n ( g 0 , ξ ) , (1.6) where g 1 , g 0 b e linearly independen t solutions of (1.4), P n ( g , ξ ) = Z ξ 0 S n ( g , ξ ) e 2 π ir ( τ − ξ ) θ ′ (0) − s 1 − ... − s n θ ( u 1 ) s 1 ...θ ( u n ) s n θ ( ξ ) − s 1 − ... − s n θ ( ξ − u 1 ) s 1 ...θ ( ξ − u n ) s n dξ , (1.7) and S n ( g , ξ ) = X 1 ≤ α ≤ n θ ( u α ) θ ( ξ − u α − η ) θ ( u α + η ) θ ( ξ − u α ) g u α − ( s 1 + ... + s n ) θ ′ (0) θ ( ξ − η ) θ ( η ) θ ( ξ ) g . (1.8) 4 W e call them el liptic pse udop otential of defe ct 0 . Suc h pseudop oten tia ls define in tegr a ble sys- tems of the form (1.1) with m = l = n + 1 . In the case s 1 = ... = s n = r = 0 , η 0 → 0 our pseudop o ten tials coincide with elliptic pseudop o ten tials constructed in [6]. In Section 4 for k < n w e construct pseudop otentials of defe ct k . These pseudop oten t ials define systems (1.1) with m = n + 1 , l = n + k + 1 . A sp ecial class of solutions for integrable systems (1 .1) dep ending on sev eral arbitrar y functions of one v ariable can b e constructed by t he metho d of hy dro dynamic reductions [9, 1 0]. The h ydro dynamic reductions are defined b y pairs of in tegra ble compatible (1 + 1)-dimensional h ydro dynamic ty p e systems of the form r i t = v i ( r 1 , ..., r N ) r i x , i = 1 , 2 , ..., N . (1.9) These integrable systems also are of in terest themselv es. A general theory of suc h t yp e in te- grable sys tems w as dev elop ed in [11, 12]. Section 5 is dev oted to hy dro dynamic reductions of systems (1 .1) constructed in Sections 3,4. In the case k = 0 the corresp onding sys tems (1.9) are defined b y r i t = S n ( g 1 ( u ) , ξ i ) S n ( g 2 ( u ) , ξ i ) r i x , (1.10) where g 1 , g 2 are linearly indep enden t solutions of ( 1.4). F or the pseudop oten tials of defect k > 0 the corresp onding formula is similar. The functions τ ( r 1 , ..., r N ), ξ i ( r 1 , ..., r N ) , u i ( r 1 , ..., r N ) are defined by the following univ ersal ov erdetermined compatible system of PDEs of the Gibb ons- Tsarev t yp e [9, 13]: ∂ α ξ β = 1 2 π i  ρ ( ξ α − ξ β ) − ρ ( ξ α )  ∂ α τ , ∂ α = ∂ ∂ r α , (1.11) ∂ α ∂ β τ = − 1 π i ρ ′ ( ξ α − ξ β ) ∂ α τ ∂ β τ , (1.12) and ∂ α u β = 1 2 π i  ρ ( ξ α − u β ) − ρ ( ξ α )  ∂ α τ , α = 1 , ..., N , β = 1 , ..., n. (1.13) Recall that here τ is the second argumen t of the function ρ . It would b e in teresting to compare form ula s (1.11), (1.12) with form ulas (3.23)-(3.26) fro m [14]. It is easy to verify t hat the system ( 1 .11)-(1.13) is consisten t. Therefore our ( 1+1)- dimensional systems (1.1 0) admit a lo cal par a meterization by 2 N arbitrary functions of one v ariable. F or some very sp ecial v alues of parameters s α in (1.4) our systems ( 1 .10) a re relat ed to the Whitham hierarc hies [6], to the F rob enious manifolds [7 , 15], and t o the asso ciativit y equation [7, 15]. 5 2 Elliptic h yp erge ometric func t ions Define a function θ in tw o v a riables z , τ b y (1.5). W e assume that Im τ > 0. The function θ is called theta-function of or der one in one v a riable. Recall the following useful form ulas: θ ( z + 1) = θ ( z ) θ ( z + τ ) = − e − 2 π iz θ ( z ) , θ ( − z ) = − e − 2 π iz θ ( z ) , θ τ ( z ) = 1 4 π i θ ′′ ( z ) − 1 2 θ ′ ( z ) . The f o llo wing statemen ts can b e verifie d straigh tforw a r dly . Prop osition 1. The system of linear equations (1.4) is compatible for any constan ts s 1 , . . . , s n , r, η 0 . The dimension of the linear space H of solutions for system (1.4) is equal to n + 1.  Remark 1. So me co efficien ts of (1 .4) can b e written in a factorized form using the fo llo wing iden tity ρ ( u β − u α ) + ρ ( u α + η ) − ρ ( u β ) − ρ ( η ) = − θ ′ (0) θ ( u α − u β + η ) θ ( u α ) θ ( u β + η ) θ ( u α − u β ) θ ( u α + η ) θ ( u β ) θ ( η ) . W e call elemen ts of H el li ptic hyp er ge ometric functions. Prop osition 2. D efine a function F ( u 1 , ..., u n , τ ) b y the follo wing in tegral represen tation F ( u 1 , ..., u n , τ ) = Z 1 0 θ ( u 1 − t ) s 1 ...θ ( u n − t ) s n θ ′ (0) s 1 + ... + s n +1 θ ( t + η ) θ ( u 1 ) s 1 ...θ ( u n ) s n θ ( t ) s 1 + ... + s n +1 θ ( η ) e 2 π i ( s 1 + ... + s n + r ) t dt. Then the function F satisfies sys tem (1.4). Prop osition 3. Let H = H s 1 ,...,s n ,r,η 0 and e H = H s 1 ,...,s n , 0 ,r,η 0 . Then e H is spanned b y H and b y the function Z ( u 1 , ..., u n , u n +1 , τ ) = Z u n +1 0 θ ( u 1 − t ) s 1 ...θ ( u n − t ) s n θ ′ (0) s 1 + ... + s n +1 θ ( t + η ) θ ( u 1 ) s 1 ...θ ( u n ) s n θ ( t ) s 1 + ... + s n +1 θ ( η ) e 2 π i ( s 1 + ... + s n + r ) t dt. (2.14) Moreo ve r, the space H s 1 ,...,s n , 0 ,..., 0 ,r , η 0 ( m zeros) is spanned b y H and b y Z ( u 1 , ..., u n , u n +1 , τ ), Z ( u 1 , ..., u n , u n +2 , τ ) , ..., Z ( u 1 , ..., u n , u n + m , τ ).  In the simplest case n = 1 system (1.4) fo r a function g ( u 1 , τ ) ha s the followin g form g u 1 u 1 =  ( s 1 + 1) θ ′ ( u 1 + η ) θ ( u 1 + η ) + s 1 θ ′ ( − η ) θ ( − η ) − ( 2 s 1 + 1) θ ′ ( u 1 ) θ ( u 1 ) + 2 π ir  g u 1 − s 2 1 θ ′ (0) 2 θ ( u 1 − η ) θ ( u 1 + η ) θ ( u 1 ) 2 θ ( − η ) θ ( η ) g , g τ = 1 2 π i  θ ′ ( u 1 + η ) θ ( u 1 + η ) − θ ′ ( η ) θ ( η )  g u 1 + s 1 2 π i (ln θ ( η )) ′′ g . 6 If s 1 = 0, then the functions g = 1 and g = Z ( u 1 , τ ) = Z u 1 0 θ ′ (0) θ ( t + η ) θ ( t ) θ ( η ) e 2 π irt dt span the space of solutions. Moreo ver, Propo sition 3 implies that the functions 1 , Z ( u 1 , τ ), ..., Z ( u n , τ ) span the space of solutions of ( 1.4) in the case s 1 = · · · = s n = 0. 3 Elliptic ps eudop ot en tials of defect 0 F or a ny elliptic hy p ergeometric function g ∈ H we put S n ( g , ξ ) = X 1 ≤ α ≤ n θ ( u α ) θ ( ξ − u α − η ) θ ( u α + η ) θ ( ξ − u α ) g u α − ( s 1 + ... + s n ) θ ′ (0) θ ( ξ − η ) θ ( η ) θ ( ξ ) g . (3.15) Define P n ( g , ξ ) b y the formula (1.7 ) if Re ( s 1 + ... + s n ) > 1 and as the a nalytic contin uation of (1.7) otherwise. Prop osition 4. The following relations hold ( P n ( g , ξ )) u α = − g u α θ ( u α ) θ ( ξ − u α − η ) θ ( u α + η ) θ ( ξ − u α ) e 2 π ir ( τ − ξ ) θ ′ (0) − s 1 − ... − s n θ ( u 1 ) s 1 ...θ ( u n ) s n θ ( ξ ) − s 1 − ... − s n θ ( ξ − u 1 ) s 1 ...θ ( ξ − u n ) s n , (3.16) ( P n ( g , ξ )) τ =  1 2 π i ( X 1 ≤ α ≤ n θ ( u α ) θ ′ ( ξ − u α − η ) θ ( u α + η ) θ ( ξ − u α ) g u α − ( s 1 + ... + s n ) θ ′ (0) θ ′ ( ξ − η ) θ ( η ) θ ( ξ ) g ) − θ ′ ( − η ) 2 π iθ ( − η ) ( X 1 ≤ α ≤ n θ ( u α ) θ ( ξ − u α − η ) θ ( u α + η ) θ ( ξ − u α ) g u α − ( s 1 + ... + s n ) θ ′ (0) θ ( ξ − η ) θ ( η ) θ ( ξ ) g )  × e 2 π ir ( τ − ξ ) θ ′ (0) − s 1 − ... − s n θ ( u 1 ) s 1 ...θ ( u n ) s n θ ( ξ ) − s 1 − ... − s n θ ( ξ − u 1 ) s 1 ...θ ( ξ − u n ) s n . (3.17) Pro of. T aking the deriv ativ es of (3.16), (3.17) with resp ect to ξ , one a r riv es at theta- functional iden tities, w hic h can b e prov ed straigh tforw ar dly . Moreov er, the v alues of the left and the right hand sides of (3.16) and (3.17) are equal to zero at ξ = 0.  Let g 1 , g 0 b e linearly indep enden t eleme n ts of H . A pseudop o ten tial A n ( p, u 1 , ..., u n , τ ) defined in a pa r ametric form b y (1.6 ) is called el liptic pse udop otential of de fe ct 0 . Relations (1.6) mean tha t to find A n ( p, u 1 , ..., u n , τ ) , one has to express ξ fr o m the second equation and substitute the r esult into the first equation. Let g 0 , g 1 , ..., g n ∈ H b e a basis in H . D efine pseudop oten tials B α ( p, u 1 , ..., u n , τ ) of defect 0, where α = 1 , ..., n, b y B α = P n ( g α , ξ ) , p = P n ( g 0 , ξ ) , α = 1 , ..., n. (3.18) 7 Supp ose that u 1 , ..., u n , τ ar e functions of t 0 = x, t 1 , ..., t n . Theorem 1. The compatibilit y conditions ψ t α t β = ψ t β t α for the system ψ t α = B α ( ψ x , u 1 , ..., u n , τ ) , α = 1 , ..., n, (3.19) are equiv alen t to the follo wing system of PDEs for u 1 , ..., u n , τ : X 1 ≤ β ≤ n ( g q g r,u β − g r g q ,u β )( u β ,t s + 1 2 π i ( θ ′ ( u β + η ) θ ( u β + η ) − θ ′ ( η ) θ ( η ) ) τ t s )+ X 1 ≤ β ≤ n ( g r g s,u β − g s g r,u β )( u β ,t q + 1 2 π i ( θ ′ ( u β + η ) θ ( u β + η ) − θ ′ ( η ) θ ( η ) ) τ t q ) + (3.20) X 1 ≤ β ≤ n ( g s g q ,u β − g q g s,u β )( u β ,t r + 1 2 π i ( θ ′ ( u β + η ) θ ( u β + η ) − θ ′ ( η ) θ ( η ) ) τ t r ) = 0 , X 1 ≤ β ≤ n,β 6 = α θ ( u β ) θ ( u α − u β − η ) θ ( u β + η ) θ ( u α − u β ) ( g r,u α g q ,u β − g q ,u α g r,u β )( u α,t s − u β ,t s + 1 2 π i ( θ ′ ( u α − u β − η ) θ ( u α − u β − η ) − θ ′ ( − η ) θ ( − η ) ) τ t s )+ X 1 ≤ β ≤ n,β 6 = α θ ( u β ) θ ( u α − u β − η ) θ ( u β + η ) θ ( u α − u β ) ( g s,u α g r,u β − g r,u α g s,u β )( u α,t q − u β ,t q + 1 2 π i ( θ ′ ( u α − u β − η ) θ ( u α − u β − η ) − θ ′ ( − η ) θ ( − η ) ) τ t q )+ X 1 ≤ β ≤ n,β 6 = α θ ( u β ) θ ( u α − u β − η ) θ ( u β + η ) θ ( u α − u β ) ( g q ,u α g s,u β − g s,u α g q ,u β )( u α,t r − u β ,t r + 1 2 π i ( θ ′ ( u α − u β − η ) θ ( u α − u β − η ) − θ ′ ( − η ) θ ( − η ) ) τ t r ) − ( s 1 + ... + s n ) θ ′ (0) θ ( u α − η ) θ ( η ) θ ( u α ) ( g q g r,u α − g r g q ,u α )( u α,t s + 1 2 π i ( θ ′ ( u α − η ) θ ( u α − η ) − θ ′ ( − η ) θ ( − η ) ) τ t s ) − ( s 1 + ... + s n ) θ ′ (0) θ ( u α − η ) θ ( η ) θ ( u α ) ( g r g s,u α − g s g r,u α )( u α,t q + 1 2 π i ( θ ′ ( u α − η ) θ ( u α − η ) − θ ′ ( − η ) θ ( − η ) ) τ t q ) − (3.21) ( s 1 + ... + s n ) θ ′ (0) θ ( u α − η ) θ ( η ) θ ( u α ) ( g s g q ,u α − g q g s,u α )( u α,t r + 1 2 π i ( θ ′ ( u α − η ) θ ( u α − η ) − θ ′ ( − η ) θ ( − η ) ) τ t r ) = 0 , where α = 1 , ..., n . Here q , r , s r un fro m 0 to n . Pro of. T aking in to accoun t (3.18), we find that the compatibilit y conditions for (3.19) are equiv a len t to n X α =1  (( P n ( g q , ξ )) ξ ( P n ( g r , ξ )) u α − ( P n ( g r , ξ )) ξ ( P n ( g q , ξ )) u α ) u α,t s + (( P n ( g r , ξ )) ξ ( P n ( g s , ξ )) u α − ( P n ( g s , ξ )) ξ ( P n ( g r , ξ )) u α ) u α,t q + (( P n ( g s , ξ )) ξ ( P n ( g q , ξ )) u α − ( P n ( g q , ξ )) ξ ( P n ( g s , ξ )) u α ) u α,t r  + (3.22) (( P n ( g q , ξ )) ξ ( P n ( g r , ξ )) τ − ( P n ( g r , ξ )) ξ ( P n ( g q , ξ )) τ ) τ t s + 8 (( P n ( g r , ξ )) ξ ( P n ( g s , ξ )) τ − ( P n ( g s , ξ )) ξ ( P n ( g r , ξ )) τ ) τ t q + (( P n ( g s , ξ )) ξ ( P n ( g q , ξ )) τ − ( P n ( g q , ξ )) ξ ( P n ( g s , ξ )) τ ) τ t r = 0 . Using (1.7), (3 .17), we rewrite (3.22) as follo ws: X 1 ≤ β ≤ n θ ( u β ) θ ( ξ − u β − η ) θ ( u β + η ) θ ( ξ − u β ) ( S n ( g q , ξ ) g r,u β − S n ( g r , ξ ) g q ,u β ) u β ,t s − 1 2 π i X 1 ≤ β ≤ n θ ( u β ) θ ′ ( ξ − u β − η ) θ ( u β + η ) θ ( ξ − u β ) ( S n ( g q , ξ ) g r,u β − S n ( g r , ξ ) g q ,u β ) τ t s + (3.23) 1 2 π i ( s 1 + ... + s n ) θ ′ (0) θ ′ ( ξ − η ) θ ( η ) θ ( ξ ) ( S n ( g q , ξ ) g r − S n ( g r , ξ ) g q ) τ t s + ( q , r, s ) = 0 , where ( q , r , s ) means the cyclic p erm utation of q , r , s . Denote the left hand side of (3.23) by Λ( ξ ). One can c hec k that Λ( ξ + 1) = Λ( ξ ) , Λ( ξ + τ ) = e 4 π iη Λ( ξ ) , and the only singularities of Λ( ξ ) are p oles of order one at the p oints ξ = 0 , u 1 , ..., u n mo dulo 1 , τ . This implies that Λ( ξ ) = 0 iff the residues at these p oints are equal to zero. Calculating the residue at ξ = 0, we get (3 .2 0). The calculation of the residue at ξ = u α leads to (3.21).  Remark 2. Given t 1 , t 2 , t 3 , Theorem 1 yields a 3-dimensional system of the form (1.1) with l = m = n + 1 p ossessing a pseudop oten tial represen t a tion. Remark 3. Conside r the case s 1 = ... = s n = 0 . W e ha v e g = c 0 + c 1 Z ( u 1 , τ ) + ... + c n Z ( u n , τ ) , where c 0 , ..., c n are constan ts. Therefore, S n ( g , ξ ) = X 1 ≤ α ≤ n c α e 2 π iru α θ ′ (0) θ ( ξ − u α − η 0 ) θ ( η 0 ) θ ( ξ − u α ) . If we a ssume r = 0 and c 1 + ... + c n = 0, then in the limit η 0 → 0 w e obtain S n ( g , ξ ) = X 1 ≤ α ≤ n c α ρ ( ξ − u α ) . A system of PD Es equiv alen t to compatibilit y conditions for equations of the form (3.22), was called in [6] a Whitham hier ar chy . In this pap er I.M. Kric hev er c onstructed some Whitham hierarc hies related to algebraic curv es of arbitrar y g enus g . The hierarch y corresponding to g = 1 is equiv alen t to one describ ed by Theorem 1 if r = s 1 = . . . = s n = 0 , c 1 + .. + c n = 0 , and η 0 → 0 as describ ed ab o v e. 9 4 Elliptic ps eudop ot en tials of defect k > 0 In t his section w e construct el liptic pseudop otentials of defe ct k. Fix k linearly independen t elliptic h yp ergeometric functions h 1 , ..., h k ∈ H . F or a ny g ∈ H define P n,k ( g , ξ ) b y the formula P n,k ( g , ξ ) = 1 ∆ det     P n ( g , ξ ) P n ( h 1 , ξ ) ... P n ( h k , ξ ) g u n − k +1 h 1 ,u n − k +1 ... h k ,u n − k +1 ......... ... ... ......... g u n h 1 ,u n ... h k ,u n     . (4.24) Here ∆ = det   h 1 ,u n − k +1 ... h k ,u n − k +1 ......... ... ......... h 1 ,u n ... h k ,u n   and P n ( g , ξ ) is given b y (1 .7). Notice that P n,k ( h 1 , ξ ) = ... = P n,k ( h k , ξ ) = 0. It is easy to see that linear tr ansformations of the form h i → c i 1 h 1 + ... + c ik h k , g → g + d 1 h 1 + ... + d k h k with constan t co efficien ts c ij , d i do not change P n,k ( g , ξ ). One can verify that ( P n,k ( g , ξ )) ξ = S n,k ( g , ξ ) e 2 π ir ( τ − ξ ) θ ′ (0) − s 1 − ... − s n θ ( u 1 ) s 1 ...θ ( u n ) s n θ ( ξ ) − s 1 − ... − s n θ ( ξ − u 1 ) s 1 ...θ ( ξ − u n ) s n , (4.25) where S n,k ( g , ξ ) = 1 ∆ ( X 1 ≤ α ≤ n − k θ ( u α ) θ ( ξ − u α − η ) θ ( u α + η ) θ ( ξ − u α ) ∆ α ( g ) − ( s 1 + ... + s n ) θ ′ (0) θ ( ξ − η ) θ ( η ) θ ( ξ ) ∆ 0 ( g )) (4.26) and ∆ α ( g ) = det     g u α h 1 ,u α ... h k ,u α g u n − k +1 h 1 ,u n − k +1 ... h k ,u n − k +1 ......... ... ... ......... g u n h 1 ,u n ... h k ,u n     , ∆ 0 ( g ) = det     g h 1 ... h k g u n − k +1 h 1 ,u n − k +1 ... h k ,u n − k +1 ......... ... ... ......... g u n h 1 ,u n ... h k ,u n     . Prop osition 5. The following relations hold ( P n,k ( g , ξ )) u α = − ∆ α ( g ) θ ( u α ) ∆ θ ( u α + η ) X n − k +1 ≤ β ≤ n θ ( u β − u α − η )( P n,k ( g , ξ )) u β θ ( u β − u α ) S n,k ( g , u β ) − (4.27) ∆ α ( g ) θ ( u α ) θ ( ξ − u α − η ) ∆ θ ( u α + η ) θ ( ξ − u α ) e 2 π ir ( τ − ξ ) θ ′ (0) − s 1 − ... − s n θ ( u 1 ) s 1 ...θ ( u n ) s n θ ( ξ ) − s 1 − ... − s n θ ( ξ − u 1 ) s 1 ...θ ( ξ − u n ) s n , 10 where 1 ≤ α ≤ n − k , and ( P n,k ( g , ξ )) τ = 1 2 π i X n − k +1 ≤ β ≤ n ( P n,k ( g , ξ )) u β S n,k ( g , u β ) ( S ′ n,k ( g , u β ) − θ ′ ( − η ) θ ( − η ) S n,k ( g , u β )) + (4.28) 1 2 π i  S ′ n,k ( g , ξ ) − θ ′ ( − η ) θ ( − η ) S n,k ( g , ξ )  e 2 π ir ( τ − ξ ) θ ′ (0) − s 1 − ... − s n θ ( u 1 ) s 1 ...θ ( u n ) s n θ ( ξ ) − s 1 − ... − s n θ ( ξ − u 1 ) s 1 ...θ ( ξ − u n ) s n , where S ′ n,k ( g , ξ ) = 1 ∆ ( X 1 ≤ α ≤ n − k θ ( u α ) θ ′ ( ξ − u α − η ) θ ( u α + η ) θ ( ξ − u α ) ∆ α ( g ) − ( s 1 + ... + s n ) θ ′ (0) θ ′ ( ξ − η ) θ ( η ) θ ( ξ ) ∆ 0 ( g )) . Moreo ve r, ( P n,k ( g , ξ )) u β S n,k ( g , u β ) do es no t dep end on g if n − k + 1 ≤ β ≤ n . Pro of. T aking the deriv ativ es of (4.27), (4.28) with resp ect to ξ , one a r riv es at theta- functional iden tities, w hic h can b e prov ed straigh tforw ar dly . Moreov er, the v alues of the left and the right hand sides of (4.27) and (4.28) are equal to zero at ξ = 0.  Let g 1 , g 2 ∈ H . Ass ume that g 1 , g 2 , h 1 , ..., h k are linearly indep enden t. D efine pseudopo ten- tial A n,k ( p, u 1 , ..., u n , τ ) in the parametric for m b y A n,k = P n,k ( g 1 , ξ ) , p = P n,k ( g 2 , ξ ) . (4.29) T o construct A n,k ( p, u 1 , ..., u n , τ ), one has to find ξ from the second e quation and s ubstitute in to the first one. The pseudop oten tia l A n,k ( p, u 1 , ..., u n , τ ) is called e l liptic pseudop otential of defe ct k . Theorem 2. Let g 0 , g 1 , ..., g n − k , h 1 , ..., h k ∈ H b e a basis in H and pseudopo ten tials B α , α = 1 , ..., n − k ar e defined b y B α = P n,k ( g α , ξ ) , p = P n,k ( g 0 , ξ ) , α = 1 , ..., n − k . Then the compatibility conditions fo r (3.19) are equiv alen t to the follo wing system of PDEs for u 1 , ..., u n , τ : X 1 ≤ β ≤ n − k (∆ 0 ( g q )∆ β ( g r ) − ∆ 0 ( g r )∆ β ( g q ))( u β ,t s + 1 2 π i ( θ ′ ( u β + η ) θ ( u β + η ) − θ ′ ( η ) θ ( η ) ) τ t s )+ X 1 ≤ β ≤ n − k (∆ 0 ( g r )∆ β ( g s ) − ∆ 0 ( g s )∆ β ( g r ))( u β ,t q + 1 2 π i ( θ ′ ( u β + η ) θ ( u β + η ) − θ ′ ( η ) θ ( η ) ) τ t q ) + (4.30) X 1 ≤ β ≤ n − k (∆ 0 ( g s )∆ β ( g q ) − ∆ 0 ( g q )∆ β ( g s ))( u β ,t r + 1 2 π i ( θ ′ ( u β + η ) θ ( u β + η ) − θ ′ ( η ) θ ( η ) ) τ t r ) = 0 , X 1 ≤ β ≤ n − k ,β 6 = α θ ( u β ) θ ( u α − u β − η ) θ ( u β + η ) θ ( u α − u β ) (∆ α ( g r )∆ β ( g q ) − ∆ α ( g q )∆ β ( g r )) × 11 ( u α,t s − u β ,t s + 1 2 π i ( θ ′ ( u α − u β − η ) θ ( u α − u β − η ) − θ ′ ( − η ) θ ( − η ) ) τ t s )+ X 1 ≤ β ≤ n − k ,β 6 = α θ ( u β ) θ ( u α − u β − η ) θ ( u β + η ) θ ( u α − u β ) (∆ α ( g s )∆ β ( g r ) − ∆ α ( g r )∆ β ( g s )) × ( u α,t q − u β ,t q + 1 2 π i ( θ ′ ( u α − u β − η ) θ ( u α − u β − η ) − θ ′ ( − η ) θ ( − η ) ) τ t q )+ X 1 ≤ β ≤ n − k ,β 6 = α θ ( u β ) θ ( u α − u β − η ) θ ( u β + η ) θ ( u α − u β ) (∆ α ( g q )∆ β ( g s ) − ∆ α ( g s )∆ β ( g q )) × ( u α,t r − u β ,t r + 1 2 π i ( θ ′ ( u α − u β − η ) θ ( u α − u β − η ) − θ ′ ( − η ) θ ( − η ) ) τ t r ) − ( s 1 + ... + s n ) θ ′ (0) θ ( u α − η ) θ ( η ) θ ( u α ) (∆ 0 ( g q )∆ α ( g r ) − ∆ 0 ( g r )∆ α ( g q ))( u α,t s + 1 2 π i ( θ ′ ( u α − η ) θ ( u α − η ) − θ ′ ( − η ) θ ( − η ) ) τ t s ) − ( s 1 + ... + s n ) θ ′ (0) θ ( u α − η ) θ ( η ) θ ( u α ) (∆ 0 ( g r )∆ α ( g s ) − ∆ 0 ( g s )∆ α ( g r ))( u α,t q + 1 2 π i ( θ ′ ( u α − η ) θ ( u α − η ) − θ ′ ( − η ) θ ( − η ) ) τ t q ) − (4.31) ( s 1 + ... + s n ) θ ′ (0) θ ( u α − η ) θ ( η ) θ ( u α ) (∆ 0 ( g s )∆ α ( g q ) − ∆ 0 ( g q )∆ α ( g s ))( u α,t r + 1 2 π i ( θ ′ ( u α − η ) θ ( u α − η ) − θ ′ ( − η ) θ ( − η ) ) τ t r ) = 0 , where α = 1 , ..., n − k and n − k X α =1 ∆ α ( g r ) θ ( u α ) θ ( u β − u α − η ) ∆ θ ( u α + η ) θ ( u β − u α ) u α,t s − S n,k ( g r , u β ) u β ,t s − 1 2 π i ( S ′ n,k ( g r , u β ) − θ ′ ( − η ) θ ( − η ) S n,k ( g r , u β )) τ t s = (4.32) n − k X α =1 ∆ α ( g s ) θ ( u α ) θ ( u β − u α − η ) ∆ θ ( u α + η ) θ ( u β − u α ) u α,t r − S n,k ( g s , u β ) u β ,t r − 1 2 π i ( S ′ n,k ( g s , u β ) − θ ′ ( − η ) θ ( − η ) S n,k ( g s , u β )) τ t r , where β = n − k + 1 , ..., n . Here q , r, s run f rom 0 to n and t 0 = x . Pro of is similar to the pro of of Theorem 1.  Remark 4. Given t 1 , t 2 , t 3 , Theorem 2 yields a 3-dimensional system of the form (1.1) with m = n + 1 , l = n + k + 1 p ossessing a pseudop oten tial represen ta t io n. Indeed, form ula (4.32) giv es 3 k linearly indep enden t equations if q , r , s = 1 , 2 , 3. F orm ulas (4.30 ) , (4.31) giv e n − k + 1 equations. On the other hand, one can construct exactly k linear com binations of equations (4.32) with q , r, s = 1 , 2 , 3 suc h that deriv ativ es of u i , i = n − k + 1 , ..., n cancel out. Moreo v er, these linear com binations b elong to the span of equations (4.30), (4.31). T herefore there exist ( n − k + 1) + 3 k − k = n + k + 1 linearly indep enden t equations. Remark 5 . W e hav e to assume n ≥ k + 2 in (4.30 ) , (4.31), (4.32). Indeed, for n = k + 1 w e cannot c onstruct more t hen one pse udop oten tial and t herefore there is no any system of the form (1 .1) asso ciated with this case. How ev er, the corresp onding pseudop oten tial generates 12 in teresting integrable (1+1 )-dimensional systems of h ydro dynamic type (see Section 5). Prob- ably these pseudopotentials for k = 0 , 1 , ... a re also related to some infinite in tegrable c hains o f the Benney type [16, 17]. System (4.30)-(4.32) p ossesse s many conserv ation laws of the h ydro dynamic t yp e. In par- ticular, the follo wing statemen t can b e v erified b y a straightforw ard calculation. Prop osition 6. F or any r 6 = s = 0 , 1 , ..., n, system (4.30)-(4.32) has k conserv ation la ws of the form: ∆( g r , h 1 , ... ˆ i...h k ) ∆( h 1 , ..., h k ) ! t s = ∆( g s , h 1 , ... ˆ i...h k ) ∆( h 1 , ..., h k ) ! t r , (4.33) where i = 1 , ..., k . Here ∆( f 1 , ..., f k ) = det   f 1 ,u n − k +1 ... f k ,u n − k +1 ......... ... ......... f 1 ,u n ... f k ,u n   .  Prop osition 6 allows us to define functions z 1 , ..., z k suc h that ∆( g r , h 1 , ... ˆ i...h k ) ∆( h 1 , ..., h k ) = z i,t r (4.34) for all i = 1 , ..., k and r = 0 , 1 , ..., n . Supp ose n + 1 ≥ 3 k ; then the system of the form (1.1) obtained from ( 4 .30)-(4.32) with q , r , s = 1 , 2 , 3 consists of 3 k equations (4.32) (they are equiv alen t to (4.3 3)) and n + 1 − 2 k equations o f the form (4 .3 0), (4.31). Indeed, o nly n + 1 − 2 k equations (4 .30), (4.31) ar e linearly indep enden t from (4.3 2). Expressing τ , u 1 , ..., u 3 k − 1 in terms of z i,t 1 , z i,t 2 , z i,t 3 , i = 1 , ..., k fro m (4.34) and su bstituting in to n + 1 − 2 k equations of the form (4.3 0), (4.31), we obtain a 3- dimensional system of n + 1 − 2 k equations for n + 1 − 2 k unknow ns z 1 , ..., z k , u 3 k , ...u n . This is a quasi-linear system of the second order with respect to z i and of the first order with respect to u j , whose co efficien ts de p end on z i,t 1 , z i,t 2 , z i,t 3 , i = 1 , ..., k , and u 3 k , ...u n . It is clear that the general solutio n of the system can b e lo cally pa rameterized b y n + 1 − k functions in t wo v ariables. In the case 2 k ≤ n + 1 < 3 k the functions z i,t 1 , z i,t 2 , z i,t 3 , i = 1 , ..., k are functionally dep enden t. W e ha v e 3 k − n − 1 equations of the f orm R i ( z 1 ,t 1 , z 1 ,t 2 , z 1 ,t 3 , ..., z k ,t 1 , z k ,t 2 , z k ,t 3 ) = 0 , i = 1 , ..., 3 k − n − 1 and n + 1 − 2 k second order quasi-linear equations. T otally w e ha ve (3 k − n − 1) + ( n + 1 − 2 k ) = k equations f or k unknow ns z 1 , ..., z k . It is clear that the general solutio n of this system can b e lo cally parameterized b y n + 1 − k f unctions in t w o v ariables. Supp ose n + 1 < 2 k ; then we hav e n + 1 + k < 3 k , whic h means that 3 k equations of the form (4 .32) a re linearly dep enden t. Probably in this case the general solution o f the system can also b e lo cally parameterized b y n + 1 − k functions in t w o v ariables. 13 One of t he most in teresting cases is n + 1 = 3 k , when w e hav e a system o f k quasi-linear second or der equations for the functions z 1 , ..., z k . The simplest case is k = 2 . 5 In tegr abl e ( 1 +1)-dimensi o nal h ydr o dynamic-t yp e sys - tems and h ydro d yn amic re ductions In this section we presen t inte grable (1+1)-dimensional h ydro dynamic t yp e system s (1.9) con- structed in terms of elliptic h yp ergeometric functions. These systems app ear as the so-called h ydro dynamic reductions of our elliptic pseudop oten tials A n,k . Results and f orm ulas of this section lo ok similar to the ra tional case (see [1 ]). By integrabilit y of (1.9) w e mean the exis- tence of infinite n umber of hy dro dynamic commuting flo ws and conserv ation la ws. It is kno wn [12] that this is equiv alen t to the follo wing relations f or the v elo cities v i ( r 1 , ..., r N ): ∂ j ∂ i v k v i − v k = ∂ i ∂ j v k v j − v k , i 6 = j 6 = k . (5.35) Here ∂ α = ∂ ∂ r i , α = 1 , . . . , N . The system (1.9) is called semi-Ham iltonian if conditions (5.3 5) hold. The main geometrical ob ject related to any semi-Hamiltonian system (1.9 ) is a diagonal metric g k k , k = 1 , . . . , N , where 1 2 ∂ i log g k k = ∂ i v k v i − v k , i 6 = k . (5.36) In view o f (5.35), the ov erdetermined system (5.36) is compatible and the function g k k is defined up to an arbit r a ry factor η k ( r k ). The me tric g k k is called the metric asso ciate d with (1.9). It is known that tw o hydrodynamic t yp e systems are compatible iff they possess a common asso ciated metric [1 2]. A diagonal metric g k k is called a metric of Egor ov typ e if fo r any i, j ∂ i g j j = ∂ j g ii . (5.37) Note that if an Egorov-t yp e metric a sso ciated with a h ydro dynamic-t yp e system of the for m (1.9) exists, then it is unique. F or an y Egorov’s metric there exists a p oten tia l G suc h that g ii = ∂ i G . Semi-Hamiltonian systems p ossessing a ssociated metrics of Egorov type play imp ortan t role in the theory of WDV V asso ciativit y equations and in the theory of F rob enious manifolds [7, 15, 18]. Let τ ( r 1 , ..., r N ) , ξ 1 ( r 1 , ..., r N ) , ..., ξ N ( r 1 , ..., r N ) b e a solution o f the system ( 1 .11), (1.12). It can b e easily v erified that this system is in inv olution and therefore its solution admits a lo cal parameterization by 2 N functions of one v ariable. Let u 1 ( r 1 , ..., r N ) , ..., u n ( r 1 , ..., r N ) b e a solution of the system (1.13). It is easy to v erify that this system is in inv olution f or eac h fixed β and therefore has a n one-parameter family of solutions for fixed ξ i , τ . 14 Consider the follo wing system r i t = S n,k ( g 1 , ξ i ) S n,k ( g 2 , ξ i ) r i x , (5.38) where g 1 , g 2 are linearly indep enden t solutions of (1.4), the p olynomials S n,k , k > 0 are defined b y (4.26), and S n, 0 = S n (see (3.15)). Theorem 3. The system (5.38) is semi-Hamiltonian. The associat ed metric is giv en by g ii =  S n,k ( g , ξ ) e 2 π ir ( τ − ξ ) θ ′ (0) − s 1 − ... − s n θ ( u 1 ) s 1 ...θ ( u n ) s n θ ( ξ ) − s 1 − ... − s n θ ( ξ − u 1 ) s 1 ...θ ( ξ − u n ) s n  2 ∂ i τ . Pro of. Substituting the expression for the metric in to (5.36), where v i are s p ecified b y (5.38), one obtains the iden tity b y virtue of (1.4) and (1.11)-(1.13).  Remark 6. The system (5.38) do es no t p ossess the asso ciated metric of the Egorov t yp e in general. How ev er, for v ery sp ecial v alues of t he parameters s i in (1.4) there exists g 2 ∈ H suc h that the metric is of the Egorov t yp e fo r all solutions of the system (1 .11)-(1.13). Prop osition 7. Supp ose t hat a solutio n ξ 1 , ..., ξ N , τ , u 1 , ..., u n of (1 .1 1)-(1.13) is fixed. Then the hy dro dynamic t yp e systems r i t 1 = S n,k ( g 1 , ξ i ) S n,k ( g 3 , ξ i ) r i x , r i t 2 = S n,k ( g 2 , ξ i ) S n,k ( g 3 , ξ i ) r i x (5.39) are compatible for all g 1 , g 2 . Pro of. Indeed, the metric associated with (5.3 8 ) do es not depend on g 2 . Therefore the systems (5.39) has a common metric depending o n g 3 and on a solution of (1.11)-(1.13).  Remark 7. One can a lso construct some compatible sy stems of the form (5.39) us ing Prop osition 3. Set g 2 = Z ( u 1 , ..., u n , u n +1 , τ ) in (5.39). Here u n +1 is an arbitrary solution of (1.13) (with n replaced b y n + 1) distinct from u 1 , ..., u n . It is clear that the flo ws (5.39) a r e compatible for suc h g 2 and any g 1 ∈ H . Moreov er, Prop osition 3 implies that the flo ws (5.39) are compatible if w e set g 1 = Z ( u 1 , ..., u n , u n +1 , τ ), g 2 = Z ( u 1 , ..., u n , u n +2 , τ ) fo r tw o arbitrary solutions u n +1 , u n +2 of (1.13). All members of the hierarc h y constructed in Prop osition 7 p ossess a dis p ersionless Lax represen tation of the form L t = { L, A } , (5.40) where { L, A } = A p L x − A x L p , with common L = L ( p, r 1 , ..., r N ). Define a function L ( ξ , r 1 , ..., r N ) b y the follo wing syste m ∂ α L = − 1 2 π i  ρ ( ξ α − ξ ) − ρ ( ξ α )  L ξ ∂ α τ , α = 1 , ..., N . (5.41) Note that the system (5 .4 1) is in inv olution and t herefore the f unction L ( ξ , r 1 , ..., r N ) is uniquely defined up to inessen tial transformations L → λ ( L ). T o find the function L ( p, r 1 , ..., r N ) one has to express ξ in terms of p b y (1.6) for k = 0 or b y (4.29) f o r k > 0. 15 Prop osition 8. Let u 1 , . . . , u n b e arbitrary solution of (1.13 ) . Then system (5.38) admits the disp ersionless Lax represen tation (5.40), where A = A n,k is defined by (1.6) for k = 0 and b y (4.29) for k > 0. Pro of. Define A = A n,k b y (1.6) for k = 0 and b y (4 .29) for k > 0 . Substituting A into (5.40) a nd calculating L t b y virtue of (5.38), w e obtain that (5.40) is equiv alen t to ∂ i L = ∂ i P n,k ( g 2 , ξ ) · S n,k ( g 1 , ξ i ) − ∂ i P n,k ( g 1 , ξ ) · S n,k ( g 2 , ξ i ) P n,k ( g 2 , ξ ) ξ · S n,k ( g 1 , ξ i ) − P n,k ( g 1 , ξ ) ξ · S n,k ( g 2 , ξ i ) L ξ . T aking into accoun t (4.25),(5.4 1) and writing do wn P n,k ( g i , ξ ) u 1 , ..., P n,k ( g i , ξ ) u n − k and P n,k ( g i , ξ ) τ in terms of P n,k ( g 1 , ξ ) u n − k +1 , ..., P n,k ( g 1 , ξ ) u n b y ( 4 .27), (4.2 8), one can readily ve rify this equalit y .  Let us show that in tegrable (1+ 1 )-dimensional systems (5.38) define h ydro dynamic reduc- tions for pseudop oten tials and 3-dimensional systems from Sections 3 and 4. In [10, 13, 1 9] a definition of in tegrabilit y for equations (5.40), (1.2) and ( 1 .1) is g iven in terms of hy dro dynamic reductions. Supp ose there exists a pair of compatible semi-Hamiltonian h ydro dynamic-type systems of the form r i t 1 = v i 1 ( r 1 , ..., r N ) r i x , r i t 2 = v i 2 ( r 1 , ..., r N ) r i x (5.42) and f unctions u i = u i ( r 1 , ..., r N ) such that these functions satisfy (1.1) for an y solution of (5.42). Then (5 .42) is called a hydr o dynamic r e duction for ( 1 .1). Definition 1 [10]. A system of the form (1 .1) is called inte gr able if equation (1.1 ) p ossesse s sufficien tly man y h ydro dynamic reductions for eac h N ∈ N . ”Sufficien tly many” means that the set of h ydro dynamic reductions can b e lo cally parameterized b y 2 N functions of one v ari- able. Note that due to gauge tra nsformations r i → λ i ( r i ) w e hav e only N essen t ia l functional parameters for hydrodynamic reductions. Supp ose there exists a semi-Hamiltonian hy dro dynamic-t yp e syste m (1.9) and functions u i = u i ( r 1 , ..., r N ), L = L ( p, r 1 , ..., r N ) s uc h that thes e functions s atisfy dispersionless Lax equation (5.4 0) for any solution r 1 ( x, t ) , ..., r N ( x, t ) of the system (1.9 ) . Then (1.9) is called a hydr o dynamic r e duction for (5.4 0). Definition 2 [19]. A disp ersionless Lax equation (5.40) is called in te g r a ble if equation (5 .40) p ossess es sufficien tly man y hydrodynamic reductions for eac h N ∈ N . W e also call the corresp onding pseudop oten tial A ( p, u 1 , ..., u n ) integrable. Definition 3 [1]. Tw o in t egr a ble pse udop oten tials A 1 , A 2 are called c omp atible if the system L t 1 = { L, A 1 } , L t 2 = { L, A 2 } p ossess es sufficien tly man y hydrodynamic reductions (5.42 ) for eac h N ∈ N . 16 If A 1 , A 2 are compatible, then A = c 1 A 1 + c 2 A 2 is in t egr a ble for an y constan ts c 1 , c 2 . Indeed, the system r i t = ( c 1 v i 1 ( r ) + c 2 v i 2 ( r )) r i x is a hy dro dynamic reduction of (5.40). Definition 4. By 3-dimensional system asso ciated with compatible functions A 1 , A 2 w e mean the system of the form (1.1) equiv alen t to the compat ibility conditions for t he system ψ t 2 = A 1 ( ψ t 1 , u 1 , ..., u n ) , ψ t 3 = A 2 ( ψ t 1 , u 1 , ..., u n ) . (5.43) It is clear that a ny system asso ciated with a pair of compatible functions p ossesses suffi- cien tly man y hydrodynamic reductions a nd therefore it is in tegrable in the sense o f D efinition 1. The f o llo wing statemen t is a r efo r m ulation of Prop osition 8. Theorem 4. The system (5.3 8) is a hydrodynamic reduction of the pseudop oten tial A n,k defined by (1.6) if k = 0 and b y (4.29) if k > 0. Recall that w e use the notation S n ≡ S n, 0 , A n ≡ A n, 0 , P n ≡ P n, 0 . Prop osition 9. Supp ose g 1 , g 2 , g 3 , h 1 , ..., h k ∈ H are linearly independen t. Define ps eu- dop oten tials A 1 , A 2 b y A 1 = P n,k ( g 1 , ξ ) , A 2 = P n,k ( g 2 , ξ ) , p = P n,k ( g 3 , ξ ) . Then A 1 and A 2 are compatible. Pro of. Note that the system (1 .11)-(1.13), (5.41) do es not dep end on g 1 , g 2 , g 3 and t herefore w e ha v e a family of functions L, ξ i , u i , τ giving h ydro dynamic reductions of the f orm (5.38) for b oth A 1 and A 2 . Moreo ve r, according to Prop osition 7 the systems r i t 1 = S n,k ( g 1 , ξ i ) S n,k ( g 3 , ξ i ) r i x , r i t 2 = S n,k ( g 2 , ξ i ) S n,k ( g 3 , ξ i ) r i x are compatible.  Remark 8. This result implies that 3-dimensional hyd ro dynamic t yp e systems constructed in Sections 3, 4 p ossess sufficien tly many h ydro dynamic reductions and therefore are in t egrable in the sence of Definition 1. Remark 9. Using Prop osition 3, one can construct compatible pseudop otentials A 1 and A 2 dep ending on different num b er of v ariables u i . Indeed, let g 1 , g 3 , h 1 , ..., h k ∈ H and g 2 = Z ( u 1 , ..., u n , u n +1 , τ ). Then A 2 dep ends on u 1 , ..., u n , u n +1 , τ and A 1 dep ends on u 1 , ..., u n , τ only . Ac kno wledgmen ts. Authors thank B. F eigin, I. Krichev er, M. P avlo v and V. Shramc henk o for f ruitful discussions. V.S. w as partially supp orted b y the RF BR grants 08- 01-464 and NS 3472.2008 .2. 17 References [1] A. Odes skii, V. 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