Recursion Operators in the Cotangent Covering of the Rddym Equation

arXiv:2106.06763v1 [nlin.SI] 12 Jun 2021 olclsmere,rcrinoperators. recursion symmetries, nonlocal 21-71-20034. tde n[,3.Rcrinoeaosfrsmere fEuto (1 Equation of symmetries [19]. for in operators Morozov p in Recursion symmetry O. arose its 3]. and and [1, integrable [8] in Lax equations studied is degenerate It linearly [16]. hierarchy of Alonso-Shabat class the to belongs EUSO PRTR NTECTNETCOVERING COTANGENT THE IN OPERATORS RECURSION 2010 e od n phrases. and words Key References Acknowledgments Conclusions 3. shadows of Lifts Actions operators Recursion Shadows laws Conservation equation Tangent result main Symmetries The 2. notation and Coordinates Preliminaries 1. Introduction h dy qain( equation rdDym The n[2,V vinocniee i lgtydffrn oain h sys the notation) different slightly (in considered Ovsienko V. [22], In h okIKwsprilyspotdb h FRGat18-29- Grant RFBR the by supported partially was ISK work The ahmtc ujc Classification. Subject Mathematics Abstract. prtr o ymtiso Ds sa xml,tecotange the example, equation an rdDym As 3D PDEs. the of symmetries for operators prtr r on.Teepsto nldsargru cri rigorous a includes property. exposition hereditary The found. are operators u v yt yt = 2( = u ..KRASIL I.S. edsrb eea ehdo osrcignnoa recur nonlocal constructing of method general a describe We x v u y xy u FTERDMEQUATION RDDYM THE OF xx ata ieeta qain,itgal ierydegen linearly integrable equations, differential Partial − r − u hdsesols y equation) Dym dispersionless th y v u u x ′ HHKADAM VERBOVETSKY A.M. AND SHCHIK xx yt u u xy yt , = Introduction + ) u = Contents x u u u xy x 35B06. x u v − 1 xy xy u y − − u xx u u y y o hc w uulyinverse mutually two which for u v xx xx − 2( u y eint hc the check to terion u xx teuto to equation nt 01 n S Grant RSF and 10013 2 + u a on by found was ) x oete were roperties rt equations, erate u h Mart´ınezthe xy sion tem ) , (2) (1) 10 10 10 9 9 8 8 7 7 6 5 4 3 1 ′ 2 I.S.KRASIL SHCHIK AND A.M. VERBOVETSKY which is a two-component extension of the rdDym equation (another two-component generalization of (1) was studied in [20]). Treating (2) as the Euler equation on the Virasoro algebra, its bi-Hamiltonian structure was established. System (2) may be also considered in the following way. Recall (see [12, 13]) that to any equation E = {F [u]=0} in unknowns u its cotangent equation T∗E

∗ F [u]=0, ℓF [u,p]=0

∗ is naturally associated, ℓF is the operator adjoint to the linearisation of F (here and below [u], [u,p], etc. denotes the variables together with their derivatives up to certain order). Then (2) may be understood as the cotangent equation of (1), with v instead of p, ‘deformed’ by the Lagrangian term δL with L = u u2 . δu y x System (2) is an Euler-Lagrange equations. Lagrangian deformations of some linearly degenerate equations were studied in [2]. In particular, the system

vyt = 2(vyuxx − vxuxy)+ uxvxy − uyvxx − 2(2kuxyux + (kuy + l)uxx), (3) uyt = uxuxy − uyuxx, which is considered below, is the deformation of the cotangent to rdDym equation 2 R with the deforming Lagrangian L = (kuy + l)ux, k, l ∈ . System (2) is obtained from (3) when k = 1, l = 0. We ﬁnd below recursion operators for symmetries of (3).

Remark 1 (courtesy O.I. Morozov). Actually, both parameters k and l are fake in a sense: the change of variables v 7→ v − 2kxux − ku + ly kills them and transforms System (3) to the pure cotangent equation of (1). In particular, the Ovsienko system (2) also reduces to the cotangent equation. Though the computations in the general case are not much harder that in a particular one, the result look more complicated. For that reason we set k = l = 0 everywhere below, i.e., the equation we study below will be of the form

vyt = 2(vyuxx − vxuxy)+ uxvxy − uyvxx, (4) uyt = uxuxy − uyuxx.

There exist several approaches to construct recursion operators (see, for example, [21, 25]). Our method is based on the technology described in [13] and orig- inating from [11]. It is based on geometrical interpretation of recursion operators for symmetries as B¨acklund auto-transformations of the equation tangent to the given one (cf. [17]) and essentially uses the theory of diﬀerential coverings [14].

Remark 2. Recursion operators as B¨acklund auto-transformations of the tangent equations initially aroused in M. Marvan’s paper [17], though implicitly the tangent equation as an instrument for computation of recursion operators appears in [9]. Informally, tangent and cotangent bundles to PDEs are discussed in the paper [15] by B. Kupershmidt. A detailed discussion of the tangent covering and its role in the theory of Hamiltonian (Poisson) structures can be found in [12] and especially in [10].

The necessary theoretical matters needed for computations are shortly described in Section 1, while Section 2 contains the main results. RO FOR THE 2-COMPONENT RDDYM 3

1. Preliminaries and notation The exposition here is based on [5, 13, 14]. Let π : E → M be a vector bundle over a smooth manifold M, dim M = n, ∞ rank π = m, and π∞ : J (π) → M be the bundle of its infinite jets. Consider an infinitely prolonged differential equation E ∈ J ∞(π) and assume that E = {F =0}, ∗ E where F ∈ Γ(π∞(ξ)), where ξ : N → M is another vector bundle. Any is naturally endowed with an n-dimensional Frobenius integrable distribution C (the Cartan distribution) that defines its geometry. The Cartan distribution is π∞-horizontal and, consequently, defines a flat connection in π∞ : E → M. ˜ ˜ A morphism of equations f : E → E is a smooth map, such that f∗(Cθ) ⊂ Cf(θ) for any θ ∈ E˜. A morphism is called a (differential) covering if it is a submersion and the restriction f | of its differential to any Cartan plane is an isomorphism. Two ∗ Cθ coverings fi : Ei → E, i = 1, 2, are called equivalent if there exists a diffeomorphism g : E1 → E2 that preserves the Cartan distributions and such that f1 = f2 ◦ g. Denote by F = F(E) the algebra of smooth functions on E. A (higher infinites- imal) symmetry of E is a π∞-vertical derivation S : F → F (a vector field on E) that preserves the Cartan distribution. There exists a one-to-one correspondence between symmetries and solutions of the equation

ℓE(ϕ)=0, where ℓE is the restriction of the linearisation ℓF : κ → P to E and κ denotes the ∗ E module of sections Γ(π∞(π)). A conservation law of is a π∞-horizontal form n−1 E ω ∈ Λh ( ) closed with respect to the horizontal de Rham diﬀerential dh. Trivial n−2 conservation laws are those of the form ω = dhρ, ρ ∈ Λ (E). Given a covering τ : E˜ → E, symmetries and conservation laws of E˜ are called nonlocal for E. Let S : F(E) → F(E˜) be a π∞-vertical derivation. It is called a τ-shadow if

C˜X ◦ S = S ◦ CX for any vector ﬁeld X on M. We say that a symmetry S˜ is a lift of the shadow S if S˜ = S. E Let an equation E be given by {F =0}. Then the system TE

F [u]=0, ℓF [u, q]=0 is called the tangent equation of E, while the projection t: [u, q] 7→ [u] is the tangent covering over E. It must be stressed that q is considered as an odd variable of degree 1. Sections of t that preserve the Cartan distributions are identiﬁed with symmetries of E and consequently B¨acklund auto-transformations Rτ,τ ′ of the form

W ❇ (5) ⑤⑤ ❇❇ ′ τ ⑤⑤ ❇❇τ ⑤⑤ ❇❇ ⑤~ ⑤ ❇ TE TE, where τ, τ ′ are coverings, are naturally interpreted as recursion operators for symmetries of E. To construct these operators, we use a scheme consisting of two steps. In the first step, we are looking for two-component conservation laws ω of TE linear on the fibers of the projection t. Provided such a law was found, we consider the covering τ = τω associated with it and, as the second step, try to find τω-shadows (also linear on fibers both of t and τω). If such a shadow S exists, it delivers the ′ R covering τ = τS and the desired recursion operator τω ,τS . ′ 4 I.S.KRASIL SHCHIK AND A.M. VERBOVETSKY

Finally, few words about the type of equations that are the subject of our study. E ∗ ˆ κ Assume = {F = 0}, F ∈ P , and consider the operator ℓF : P → ˆ formally n T∗E adjoint to ℓF , where •ˆ = hom(•, Λh). Then the system ∗ ℓF [u,p]=0, F [u] = 0 (6) is said to be the cotangent equation of E, while the projection t∗ : T∗E → E, (u,p) 7→ (u), is called the cotangent covering. Similar to the case of the tangent equation, the variable p is odd. System (6) is always an Euler-Lagrange equation with the Lagrangian density L = pj F j and we are interested in equations ∗ ℓFP[u,p]+ G[u,p]=0, F [u]=0, (‘Lagrangian deformations’ of (6)), such that (a) they are Lagrangian as well and (b) they possess a nontrivial recursion operator. Equation (2) is exactly of this TE T∗E ∗ type. Note that for Euler-Lagrange equations and coincide, since ℓF = ℓF in this case.

Coordinates. Let U ⊂ M be a coordinate neighborhood with local coordinates x = (x1,...,xn) and E ⊃ π−1(U) ≃ U × Rm be a trivialization of π with ﬁber- 1 m j ∞ wise coordinates u = (u ,...,u ). Then adapted coordinates uσ arise on J (π), j |σ| j σ where uσ correspond to the partial derivative ∂ u /∂x . The Cartan distribution is spanned by the total derivatives

∂ j ∂ Di = i + uσi j , ∂x ∂uσ i X while the Cartan connection takes ∂/∂x to Di. Equation E is given by the system l j E of relations F (x,...,uσ,... ) = 0, l =1,...,r. To restrict necessary objects to , one needs to choose internal coordinates on E (cf. [18]) and express these objects in terms of internal coordinates. In particular, the Cartan distribution on E is got by rewriting the operators Di in internal coordinates. The coordinate presentation of the linearisation operator acts on ϕ = (ϕ1,...,ϕm) by l ∂F j ℓF (ϕ)= j Dσ(ϕ ), σ,j ∂uσ X where Dσ is the composition of the total derivatives corresponding to the multi- index σ, while for the adjoint one we have l ∗ |σ| ∂F l ℓF (ψ)= (−1) Dσ j ψ , σ,l ∂uσ X where ψ = (ψ1,...,ψr). Thus, the tangent equation of E is l ∂F j j 1 m j qσ =0, F (x,...,uσ,... )=0, q = (q ,...,q ), σ,j ∂uσ X while while the cotangent one is given by l |σ| ∂F l j 1 r (−1) Dσ j p =0, F (x,...,uσ,... )=0, p = (p ,...,p ). σ,l ∂uσ X Without loss of generality, we can deﬁne two-component conservation laws as the forms 1 2 3 n ω = (X1 dx + X2 dx ) dx ∧···∧ dx ,D1(X2)= D2(X1), where X1 and X2 are smooth functions on E. Triviality of ω means existence of a potential Y , such that Di(Y )= Xi, i = 1, 2. RO FOR THE 2-COMPONENT RDDYM 5

A covering structure in a locally trivial vector bundle τ : W → E is determined by the system of pair-wise commuting vector ﬁelds

D˜ i = Di + Xi, i =1, . . . , n, (7) α α α where Xi = Xi ∂/∂w are τ-vertical fields and w are coordinates in the fiber of τ (nonlocal variables). The equalities [D˜ , D˜ ] = 0 amount to the fact that the P i j system α α wxi = Xi , i =1,...,n, α =1,..., rank τ, is compatible modulo E. Note that (7) allows to lift any differential operator ∆ in total derivatives from E to an operator ∆˜ on the covering equation. Let ω be a two-component conservation law like above. Then one can construct the covering τω as follows. In the case dim M =2 we set

wx1 = X1, wx2 = X2, and rank τω = 1. When dim M > 2, we introduce inﬁnite number of nonlocal variables wσ, where σ is a multi-index consisting of integers 2,...,n, |σ| ≥ 0, and set

σ Dσ(Xi), i =1, 2, wxi = σi (w , i> 2. Obviously, we obtain covering structures in both cases. Consider an m-component function S = (S1,...,Sm). It deﬁnes a τ-shadow if and only if ℓ˜E(S)=0, where ℓ˜E is the natural lift of the operator ℓE to the covering equation (see above). Let now S be a τ-shadow, where τ = τω in Diagram (5). Denote by pσ the ﬁber coordinates in the left copy of TE and byp ¯σ the same coordinates in the right one j ˜ j ˜ and setp ¯σ = Dσ(S ), where Dσ is the composition of total derivatives in W . This ′ gives the covering τ = τS. The desired recursion operator is obtained, when we set j j,σ l j σ S = Sl pσ + Sσw , l,σ σ X X j,σ j E where Sl , Sσ are smooth functions on .

2. The main result We now pass to the equation E given by (4) and choose the functions x, y, t and

uxi = ux...x, uxi,yj = ux...x y...y, uxi,yj = ux...x t...t, i ≥ 0, j> 0,

i times i times j times i times j times E and similar| for{z }v. Then the| total{z } derivative| {z } on acquire| {z } the| {z form} ∂ ∂ ∂ ∂ D = + u + u + u x ∂x xi+1 ∂u xi+1,yj ∂u xi+1,tj ∂u i,j xi xi,yj xi,tj X ∂ ∂ ∂ + v + v + v , xi+1 ∂v xi+1,yj ∂v xi+1,tj ∂v i,j xi xi,yj xi,tj X ∂ ∂ ∂ ∂ D = + u + u + Di Dj−1(U) y ∂y xi,y ∂u xi,yj+1 ∂u x t ∂u i,j xi xi,yj xi,tj X ∂ ∂ ∂ + v + v + Di Dj−1(V ) , xi,y ∂v xi,yj+1 ∂v x t ∂v i,j xi xi,yj xi,tj X ′ 6 I.S.KRASIL SHCHIK AND A.M. VERBOVETSKY

∂ ∂ ∂ ∂ D = + u + Di Dj−1(U) + u t ∂t xi,t ∂u x y ∂u xi,tj+1 ∂u i,j xi xi,yj xi,tj X ∂ ∂ ∂ + v + Di Dj−1(V ) + v , xi,t ∂v x y ∂v xi,tj+1 ∂v i,j xi xi,yj xi,tj X where V = 2(vyuxx −vxuxy)+uxvxy −uyvxx and U = uxuxy −uyuxx are right-hand sides of the ﬁrst and second equations in (4), respectively.

Symmetries. The deﬁning equation ℓE(S) = 0 for symmetries of (4) is 2 DyDt(ψ) = 2vyDx(ϕ) − 2vxDxDy(ϕ)+ vxyDx(ϕ) − vxxDy(ϕ) 2 −2uxyDx(ψ) − uyDx(ψ)+ uxDxDy(ψ)+2uxDy(ψ), (8) 2 DyDt(ϕ) = uxyDx(ϕ) − uyDx(ϕ)+ uxDxDy(ϕ) − uxxDy(ϕ). Solving System (8) for ϕ and ψ of order ≤ 3, we get the symmetries

S1 = xux − 2u, xvx +2v ,

S2(T )= T, 0 , ˙ S3(T )= Tux+ T x, T vx , 1 S (T )= T¨ x2 + T˙ (xu − u)+ Tu , (xv +2v)T˙ + T v , 4 2 x t x t S5(Y )= Yuy, Y vy ,

S6 = 0,uxxx , 1 S = 0, u2 + u u − u , 7 2 xx x xxx xxt 2 2 S8 = 0,uxuxx − uxtuxx + uxuxxx − 2uxuxxt + uxtt , 3 3 S = 0, u2 u2 − 3u u u + u2 − 3u2 u +3u u + u3 u − u , 9 2 x xx x xt xx 2 xt x xxt x xtt x xxx ttt 2 uxy − 2uyuxxy S10 = 0, 2 , uy uxyuyy − uyuxyy S11 = 0, 3 , uy S12 = 0, v , 1 S (T )= 0,Tu − T˙ x , 13 x 2 1 2 1 2 S (T )= 0, T¨ x2 + Tu2 + (T − T˙ x)u − (2u +3x)T˙ − Tu , 14 6 x 3 x 6 3 t S15(T )= 0,T , Y S16(Y )= 0, 2 . uy where Y =Y (y) and T = T (t) are arbitrary smooth functions and ‘dot’ denotes the t-derivative. These symmetries commute as follows1

[S1,S2(T )]=2S2(T ), [S1,S3(T )] = S3(T ),

[S1,S6]= −S6, [S1,S7]= −2S7, [S1,S8]= −3S8, [S1,S9]= −4S9,

[S1,S11]= S11, [S1,S13(T )] = −3S13(T ), [S1,S14(T )]=4S14(T ) − S13(T ),

[S1,S15(T )] = −2S15(T ), [S1,S16(Y )]=2S16(Y ); ... [S2(T1),S4(T2)= −S2(T˙1T2 − T1T˙2), [S2(T ),S9]= −S15(T ),

1We omit zero commutators. ‘Prime’ denotes the y-derivative. RO FOR THE 2-COMPONENT RDDYM 7

1 [S (T ),S (T )] = − S (T T˙ +2T˙ T ); 2 1 14 2 3 15 1 2 1 2 [S3(T1),S3(T2)] = S2(T˙1 − T1T˙2) − T1T˙2),

[S3(T1),S4(T2)] = S3(T˙1T2 − T˙2T1), ...... [S3(T ),S8]= S15(T ), [S3(T ),S9]=2S13(T ), 1 [S (T ),S (T )] = S (T˙ T + T T˙ ), 3 1 13 2 15 1 2 2 1 2 2 1 [S (T ),S (T )] = S (2T˙ T + T T˙ ) − S (T˙ T + T T˙ ); 3 1 14 2 3 13 1 2 1 2 15 1 2 2 1 2 ... [S4(T1),S4(T2)] = S4(T˙1T2 − T1T˙2), [S4(T ),S7]= −S15(T ), ...... [S4(T ),S8]= −2S13(T ), [S4(T ),S9]=3S13(T )+3S14(T ),

[S4(T1),S13(T2)] = −S13(2T˙1T2 + T1T˙2), [S4(T1),S14(T2)] = −S14(2T˙1T2 + T 1T˙2),

[S4(T1),S15(T2)] = S15(2T˙1T2 + T1T˙2); ′ ′ ′ ′ [S5(Y1),S5(Y2)] = S5(Y1 Y2 − Y2 Y1), [S5(Y1),S16(Y2)] = −S16(Y2 Y1 +2Y1 Y2);

[S6,S12]= S6;[S7,S12]= S7;[S8,S12]= S8;[S9,S12]= S9;

[S10,S12]= S10;[S11,S12]= S11;[S12,S13(T )] = −S13(T ),

[S12,S15(T )] = −S15(T ), [S12S16(Y )] = −S16(Y ). Tangent equation. Due to (8), the tangent equation is obtained by adding to System (4) the equations

qyt = 2vypxx − 2vxpxy + vxypx −vxxpy − 2uxyqx − uyqxx + uxqxy +2uxqy, (9)

pyt = uxypx − uypxx + uxpxy − uxxpy. Conservation laws. There exist four conservation laws on TE of order ≤ 2 and linear with respect to the variables pσ, qσ:

ω1 = (X1 dx + T1 dt) ∧ dy, ω2 = (X2 dx + T2 dt) ∧ dy,

ω3 = (X3 dx + Y3 dy) ∧ dt, ω4 = (X4 dx + Y4 dy) ∧ dt, where Ypy Y (uypx − uxpy) X1 = − 2 ,T1 = 2 ; uy uy 2 X2 = Y (vypy + uyqy),T2 = Y (vyuypx − (2vxuy − uxvy)py − uyqx + uyuxqy);

X3 = T (2uxpx − pt), Y3 = T (uypx + uxpy);

X4 = T (vxpx + uxqx + qt), Y4 = −T (vxpy +2vypx + uyqx − 2uxqy). Setting Y = T = 1 and after slight relabeling, we obtain the following nonlocal variables associated with the above conservation laws: py w1,x = 2 , uy uxpy − uypx w1,t = 2 ; (10) uy w2,x = vypy + uyqy, 2 w2,t = vyuypx + (uxvy − 2vxuy)pyuyqx + uyuxqy;

w3,x = 2uxpx − pt, w3,y = uypx + uxpy; (11) w4,x = vxpx + uxqx + qt, w4,y = 2vypx − vxpy − uyqx +2uxqy. ′ 8 I.S.KRASIL SHCHIK AND A.M. VERBOVETSKY

All the nonlocal variables wi are odd of degree 1.

Shadows. Direct computations reveal two shadows of symmetries that linearly depend on the variables pσ, qσ, and wi,σ up to second order:

s0 :p ¯0 = p, q¯0 = q;

s1 :p ¯1 = uxp − w3, q¯1 = vxp − 2uxq + w4; w2 s2 :p ¯2 = uyw1, q¯2 = vyw1 + 2 . uy

The shadow s0 is responsible for the identical operator and thus is of no interest, while the other two ones provide nontrivial results.

Recursion operators. Using the last formulas, we obtain the following expressions for the nonlocal variables wi:

p¯2 w1 = , w2 = uy(uyq¯2 − vyp¯2); (12) uy

w3 = uxp − p¯1, w4 = −vxp +2uxq +¯q1. (13) Let us now substitute these expressions to the deﬁning equations of the coverings (10), (11). As the result, we obtain two B¨acklund auto-transformations of TE:

p¯1,x = uxxp − uxpx + pt,

p¯1,y = uxyp − uypx, (14) q¯1,x = 2vxpx − uxqx − 2uxxq + vxxp + qt,

q¯1,y = 2vypx − uyqx − 2uxyq + vxyp; and 1 p¯2,x = (uxyp¯2 + py), uy 1 p¯2,t = ((uxuxy − uyuxx)¯p2 − uypx + uxpy), uy 1 1 q¯2,x = 2 ((2vyuxy + uyvxy)¯p2 +2vypy) − (2uxyq¯2 − qy), (15) uy uy uxvxy − 2vxuxy 2uxvyuxy q¯2,t = + 2 − vxx p¯2 uy uy 2vyux 2vx 2(uyuxx − uxuxy ) ux + 2 − py + q¯2 + qy − qx. uy uy uy uy Thus, relations (14) and (15) deﬁne recursion operators for symmetries of Equa- tion (4). The p-components of these operators give recursion operators for the rdDym equation (cf. [3]).

Remark 3. Though formulas (14) and (15) look diﬀerent, essentially they present the same object. Namely, if we resolve (15) with respect to the variables px, py, qx, and qy, we shall obtain relations (14) up to some relabeling. Consequently, the above presented B¨acklund transformations are mutually inverse.

Remark 4. Note also that (14), understood as a covering over TE with the nonlocal variablesp ¯ andq ¯ by the very construction is equivalent to the one with the nonlocal variables w3 and w4. Explicitly, the corresponding gauge transformation is given by formulas (13). In a similar way, (12) delivers equivalence between (15) and the covering with the variables w1 and w2. RO FOR THE 2-COMPONENT RDDYM 9

Actions. Let us indicate how the operator (14) acts on symmetries of the equation E; due to Remark 3, the action of (15) is the opposite. Note also that of (14) is deﬁned up to the image of 0, which is S2(T ). Keeping in mind these remarks, we have:

0 7→ S2(T ) 7→ S3(T ) 7→ S4(T ) 7→ ⋆

S1 7→ ⋆

S5(Y ) 7→ 0 1 S 7→ − S 7→ 2S 7→ ⋆ 11 2 10 6 S7 7→ −S8 7→ ⋆

S9 7→ ⋆

S12 7→ ⋆ 3 S (T ) 7→ −2S (T ) 7→ S (T ) − S (T ) 15 13 2 13 14 S14(T ) 7→ ⋆

S16(Y ) 7→ 0.

Here ⋆ denotes a nonlocal result.

Lifts of shadows. Note finally that any nonlocal symmetry in the coverings (10) and (11) is determined by the coefficients at ∂/∂u, ∂/∂v, ∂/∂p, ∂/∂q, ∂/∂w1, ∂/∂w2 (in the case of (10)), and ∂/∂w3, ∂/∂w4 (for (11)). Denote these coefficients by

U, V, P, Q, W1, W2, W3, W4, respectively. We state that there exist nonlocal symmetries that are the lifts of the shadows s1 and s2 to the corresponding coverings. Namely, these symmetries are

σ1 : U = uyw1, w2 V = 2 + vyw1, uy

P = w1py, pyw2 Q = w1qy +2 3 , uy

W1 = w1w1,y,

W2 =2w1,yw2 − w2,yw1 and

σ2 : U = uxp − w3,

V = −2uxq + vxp − w4,

P = pxp,

Q =2qpx + qxp,

W3 = pqt +2ptq − vxppx + uxpqx +2uxpxq,

W4 = ppt − 2uxppx.

Moreover, σi are odd vector ﬁelds of degree 1. A direct computation shows that the super-commutators [σ1, σ1] and [σ2, σ2] of these ﬁelds vanish, which means that the constructed recursion operators are hereditary, [13]. ′ 10 I.S.KRASIL SHCHIK AND A.M. VERBOVETSKY

3. Conclusions The method used here to construct recursion operators for symmetries of Equa- tion (4) seems to be of a universal nature. In particular, it allows to test rigorously the hereditary property of nonlocal recursion operators. It would be interesting to apply it to other two-component extensions of linearly degenerate equations constructed in [2], as well as to the Dunajski equation [7], etc. It would be also interesting to describe (similar to how it was done in [3]) the algebra of nonlocal symmetries (and the corresponding coverings) under the action of recursion operators described above.

Acknowledgments Computations were done using the Jets [4] and Cadabra [6, 23, 24] software. The authors are grateful to O.I. Morozov for discussion.

References

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Trapeznikov Institute of Control Sciences, 65 Profsoyuznaya street, Moscow 117997, Russia Email address: [email protected]

Independent University of Moscow, Bolshoy Vlasyevskiy Pereulok 11, Moscow, 119002, Russia Email address: [email protected]