Nonlocal symmetries of Plebański’s second heavenly equation

Aleksandra Lelito; Oleg I. Morozov

doi:10.1080/14029251.2018.1452669

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Volume 25, Issue 2, March 2018, Pages 188 - 197

Nonlocal symmetries of Plebański’s second heavenly equation

Authors

Aleksandra Lelito, Oleg I. Morozov

Faculty of Applied Mathematics, AGH University of Science and Technology, Al. Mickiewicza 30, Cracow 30-059, Poland,alelito@agh.edu.pl,morozov@agh.edu.pl

Received 20 July 2017, Accepted 7 November 2017, Available Online 6 January 2021.

DOI: 10.1080/14029251.2018.1452669 How to use a DOI?
Keywords: Plebański’s second heavenly equation; Lax pair; differential covering; nonlocal symmetries
Abstract: We study nonlocal symmetries of Plebański’s second heavenly equation in an infinite-dimensional covering associated to a Lax pair with a non-removable spectral parameter. We show that all local symmetries of the equation admit lifts to full-fledged nonlocal symmetries in the infinite-dimensional covering. Also, we find two new infinite hierarchies of commuting nonlocal symmetries in this covering and describe the structure of the Lie algebra of the obtained nonlocal symmetries.
Copyright: © 2018 The Authors. Published by Atlantis and Taylor & Francis
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. Introduction

In this paper we study nonlocal symmetries of Plebański’s second heavenly equation, [32],

uxz−uty−uxxuyy+uxy2=0.(1.1)

This equation attached considerable attention because of its importance in general relativity. In particular, the equation is a reduction of the Einstein equations that govern self-dual gravitation fields, see [12,26,30,35] and references therein. Eq. (1.1) is an example of a nonlinear integrable equation in four independent variables. Here integrability means the existence of a Lax pair, or a differential covering,

{qt=(uxy+λ)qx−uxxqy,qz=uyyqx−(uxy−λ)qy,(1.2)

with a non-removable parameter λ. Expanding the pseudopotential q into a Taylor series q=∑k=0∞λkqk yields a new covering (4.1) with pseudopotentials q_k over Eq. (1.1). The goal of this paper is to study nonlocal symmetries for Eq. (1.1) in this covering.

Infinite-dimensional Lie algebras of nonlocal symmetries are well known to play an important role in the theory of nonlinear integrable equations and provide a useful tool to study of the latter, see e.g. [5,7,16,17,20,33,34] and references therein. Eq. (1.1) has the infinite-dimensional Lie algebra 𝔰 of local contact symmetries and, as it was shown in [24], is uniquely defined by this algebra, see also [11,25] for discussion of geometric properties of Eq. (1.1) and related equations. The algebra 𝔰 is the semi-direct product s=s∞⋊s⋄, where s∞=[[s,s],[s,s]] is an infinite-dimensional ideal and s⋄ is a three-dimensional solvable Lie algebra. We show that all local symmetries of Eq. (1.1) have lifts to nonlocal symmetries, that is to symmetries of the system (1.1),(4.1). Note that not every integrable equation allows lifts of local symmetries to nonlocal symmetries, see e.g. [10]. Also, we find two infinite hierarchies 𝔞 and 𝔟 of nonlocal symmetries such that

[a,a]=0, [b,b]=0.(1.3)

Note that existence of an infinite hierarchy of commuting flows is one of the most important properties of integrability, see [1,6,14,19,31] and references therein. Also, we find the structure of the Lie algebra^a 𝔰 ⊕ 𝔞 ⊕ 𝔟 (the sum as vector spaces). In paricular, we show that

[s∞,a]=[s∞,b]=0, [s⋄,a]=a, [s⋄,b]=b, [a,b]=a.(1.4)

There is a great many of works devoted to methods of studying nonlinear partial differential equations (PDEs) that admit nonlocal symmetries. For the case of potential symmetries, that is nonlocal symmetries corresponding to Abelian coverings ^b, see [8, Ch.7],[9], and references therein. In regard to applications of nonlocal symmetries in non-Abelian coverings to studying of exact solutions and other integrability properties of nonlinear PDEs, see [5,16,17,34] and references therein.

2. Preliminaries

All considerations in this paper are local. The presentation in this section closely follows [20,21], see also [22,23]. Let π:ℝn×ℝm→ℝn,π:(x1,…,xn,u1,…,um)↦(x1,…,xn) be a trivial bundle, and J^∞(π) be the bundle of its jets of the infinite order. The local coordinates on J^∞(π) are (xi,uα,uIα), where I = (i₁,…,i_n) is a multi-index, and for every local section f:ℝn→ℝn×ℝm of π the corresponding infinite jet j_∞(f) is a section j∞(f):ℝn→J∞(π) such that uIα(j∞(f))=∂#Ifα∂xI=∂i1+⋯+infα(∂x1)i1…(∂xn)in. We put uα=u(0,…,0)α. Also, in the case of n = 4, m = 1 we denote x¹ = t, x² = x, x³ = y, x⁴ = z and u(i,j,k,l)1=ut…tx…xy…yz…z with i times t, j times x, k times y, and l times z

The vector fields

Dxk=∂∂xk+∑#I≥0∑α=1muI+1kα∂∂uIα, k∈{1,…,n},

(i₁,…,i_k,…,i_n) + 1_k = (i₁,…,i_k + 1, …,i_n), are called total derivatives. They commute everywhere on J∞(π):[Dxi,Dxj]=0.

The evolutionary derivation associated to an arbitrary smooth function φ:J∞(π)→ℝm is the vector field

Eφ=∑#I≥0∑α=1mDI(φα)∂∂uIα,(2.1)

with DI=D(i1,…in)=Dx1i1∘⋯∘Dxnin.

A system of PDEs Fr(xi,uIα)=0,#I≤s,r∈{1,…,R}, of the order s ≥ 1 with R ≥ 1 defines the submanifold ℰ={(xi,uIα)∈J∞(π)∣DK(Fr(xi,uIα))=0,#K≥0} in J^∞(π).

A function φ:J∞(π)→ℝm is called a (generator of an infinitesimal) symmetry of ℰ when E_φ(F) = 0 on ℰ. The symmetry φ is a solution to the defining system

ℓℰ(φ)=0,(2.2)

where ℓε=ℓF∣ε with the matrix differential operator

ℓF=(∑#I≥0∂Fr∂uIαDI).

Solutions to (2.2) constitute the Lie algebra Lie (ℰ) of (infinitesimal) symmetries of equation ℰ with respect to the Jacobi bracket { φ, ψ} = E_φ(ψ) − E_ψ(φ). The subalgebra of contact symmetries of ℰ is Lie0(ℰ)=Lie(ℰ)∩C∞(J1(π),ℝm).

Denote W=ℝ∞ with coordinates ws,s∈ℕ∪{0}. Locally, an (infinite-dimensional) differential covering of ℰ is a trivial bundle τ:J∞(π)×W→J∞(π) equipped with the extended total derivatives

D˜xk=Dxk+∑s=0∞Tks(xi,uIα,wj)∂∂ws(2.3)

such that [D˜xi,D˜xj]=0 for all i ≠ j whenever (xi,uIα)∈ℰ. We define the partial derivatives of w^s by wxks=D˜xk(ws). This yields the system of covering equations

wxks=Tks(xi,uIα,wj).(2.4)

This over-determined system of PDEs is compatible whenever (xi,uIα)∈ℰ.

A covering is said to be Abelian when system (2.4) can be mapped to the form

wxk0=Tk0(xi,uIα), wxkj=Tks(xi,uIα,w0,…,wj−1), j∈ℕ

by a change of variables xi,uIα,ws. Otherwise the covering is non-Abelian.

Denote by E˜φ the result of substitution of D˜xk for Dxk in (2.1). A shadow of nonlocal symmetry of ℰ corresponding to the covering τ with the extended total derivatives (2.3), or τ-shadow, is a function φ∈C∞(ℰ×W,ℝm), such that

E˜φ(F)=0(2.5)

is a consequence of equations D_K(F) = 0 and (2.4). A nonlocal symmetry of ℰ corresponding to the covering τ (or τ -symmetry) is the vector field

E˜φ,A=E˜φ+∑s=0∞As∂∂ws,(2.6)

with As∈C∞(ℰ×W) such that φ satisfies (2.5) and

D˜xk(As)=E˜φ,A(Tks)(2.7)

for Tks from (2.3), see [10, ch. 6, §3.2].

Remark 2.1.

In general, not every τ-shadow corresponds to a τ-symmetry, since equations (2.7) provide an obstruction for existence of A^s in (2.6). But for any τ-shadow φ there exists a covering τ_φ and a nonlocal τ_φ-symmetry whose τ_φ-shadow coincides with φ, see [10, ch. 6,§5.8].

3. Local symmetries

The local contact symmetries of Eq. (1.1) are solutions φ = φ(t, x, y, z, u, u_t, u_x, u_y, u_z) to the equation

ℓℰ(φ)=DxDz(φ)−DtDy(φ)−uxxDy2(φ)+2uxyDxDy(φ)=0.

Direct computations ^c show that the Lie algebra s=Lie0(ℰ) is generated by the following functions

φ0(A)=−Azut−(xAtz+yAzz)ux+(xAtt+yAtz)uy+Atuz−16(x3Attt+3x2yAttz+3xy2Atzz+y3Azzz),φ1(A)=−Azux+Atuy−12(x2Att+2xyAtz+y2Azz),φ2(A)=−At−yAz,φ3(A)=−A,ψ1=3u−xux−yuy,ψ2=−3tut−xux−yuy−3zuz,ψ3=−tux−zuy,

where A = A(t, z) and B = B(t, z) below are arbitrary functions of t and z. The structure of s is given by equations

{φi(A),φj(B)}={φi+j(AtBz−AzBt),i+j≤4,0,i+j>4,(3.1)

{ψ1,φj(A)}=jφj(A),(3.2)

{ψ2,φj(A)}=φj(2(3−j)A−3(tAt+zAz)),(3.3)

{ψ3,φj(A)}=={φj+1((2−j)A−tAt−zAz),j≤2,0,j=3,(3.4)

{ψ1,ψ2}=0, {ψ1,ψ3}=ψ3, {ψ2,ψ3}=−2ψ3.(3.5)

Remark 3.1.

We have s=s∞⋊s⋄, where s∞ is generated by φ_i(A) and s⋄ is generated by ψ_j The algebra 𝔰_∞ admits the following description. Consider the (commutative associative) algebra of truncated polynomials ℝ₄[s] = ℝ[s]/(s)⁴ and the Lie algebra 𝔥 of Hamiltonian vector fields on ℝ² [15] Then 𝔰_∞ is isomorphic to the Lie algebra ℝ₄[s] ⊗ 𝔥 with the bracket [f ⊗ V,g ⊗ W ] = f g ⊗ [V,W ] for f ,g ∈ ℝ₄[s] and V, W ∈ 𝔥.

4. Infinite-dimensional covering, shadows, and nonlocal symmetries

Substituting

q=∑k=0∞λkqk.

in the system (1.2) yields new (infinite-dimensional) covering

{q0,t=uxyq0,x−uxxq0,y,q0,z=uyyq0,x−uxyq0,y,qm,t=uxyqm,x−uxxqm,y+qm−1,x,qm,z=uyyqm,x−uxyqm,y+qm−1,y, m≥1.(4.1)

Direct computations prove:

Proposition 4.1.

Function υ = q is a shadow in the covering (1.2).

Then we have:

Corollary 4.1.

Functions υ_k = q_k, k ≥ 0, are shadows in the covering (4.1).

A nonlocal symmetry of Eq. (1.1) is an infinite sequence (φ, Q₀, Q₁,…, Q_m,…), where φ = φ(xⁱ,u,u_xⁱ ,...,q_j,q_j,x,q_j,y,...) and Q_m = Q_m(xⁱ,u,u_xⁱ ,...,q_j,q_j,x,q_j,y,...), m ≥ 0, are solutions to the equation

φxz−φty−uyyφxx−uxxφyy+2uxyφxy=0

and the linearization

{Q0,t=uxyQ0,x+q0,xφxy−uxxQ0,y−q0,yφxx,Q0,z=uyyQ0,x+q0,xφyy−uxyQ0,y−q0,yφxy,Qm,t=uxyQm,x+qm,xφxy−uxxQm,y−qm,yφxx+Qm−1,x,Qm,z=uyyQm,x+qm,xφyy−uxyQm,y−qm,yφxy+Qm−1,y, m≥1.

of the system (4.1).

Remark 4.1.

To simplify notation, here and below we use φ_xz for D˜xD˜z(φ), etc.

The nonlocal symmetries of Eq. (1.1) are described by the following theorems:

Theorem 4.1.

The local symmetries φ₀(A), …, φ₃(A), ψ₁, ψ₂, ψ₃ have the lifts Φ₀(A), …, Φ₃(A), Ψ₁, Ψ₂, Ψ₃ to the nonlocal symmetries in the covering (4.1) defined as Φ_i(A) = (φ_i(A), Φ_i_,0(A), Φ_i_,1(A), …, Φ_i_,_k(A), …), Ψ_j = (ψ_j, Ψ_j_,0, Ψ_j_,1, …, Ψ_j_,_k, …), with

Φ0,k(A)=−Azqk,t−(xAtz+yAzz)qk,x+(xAtt+yAtz)qk,y+Atqk,z,Φ1,k(A)=−Azqk,x+Atqk,y,Φ2,k(A)=Φ3,k(A)=0,Ψ1,k=−xqk,x−yqk,y−kqk,Ψ2,k=−3tqk,t−xqk,x−yqk,y−3zqk,z+2kqk,Ψ3,k=−tqk,x−zqk,y+(k+1)qk+1.

The proof is similar to the proof of theorem 4.2 below and is therefore omitted.

Theorem 4.2.

The shadows v_k = q_k, k ≥ 0, have the lifts ϒ_k to the nonlocal symmetries in the covering (4.1) defined as ϒ_k = (q_k, ϒ_k_,0, ϒ_k_,1, … , ϒ_k_,_m, …) with

Υk,m=∑s=0m〈qs,qk+m+1−s〉,

where ⟨ a, b⟩ = a_xb_y − a_y b_x

Remark 4.2.

The bracket ⟨⋅, ⋅⟩ is a Lie bracket. It endows the space C^∞(ℝ²) of smooth functions on ℝ² with the structure of a Lie algebra with the noncentral part isomorphic to the algebra of Hamiltonian vector fields on ℝ² and the center generated by the constant functions.

Proof. We claim that^d

Υk,m,t=〈Υk,m,ux〉+〈qm,qk,x〉+Υk,m−1,x.

Indeed,

Υk,m,t=∑s=0m〈qs,qk+m+1−s〉t=∑s=0m〈qs,t,qk+m+1−s〉+∑s=0m〈qs,qk+m+1−s,t〉==〈〈q0,ux〉,qk+m+1〉+∑s=1m〈〈qs,ux〉+qs−1,x,qk+m+1−s〉++〈q0,〈qk+m+1,ux〉+qk+m,x〉+∑s=1m〈qs,〈qk+m+1−s,ux〉+qk+m−s,x〉==〈〈q0,ux〉,qk+m+1〉+∑s=1m〈〈qs,ux〉,qk+m+1−s〉+∑s=1m〈qs−1,x,qk+m+1−s〉︸s=1=∑s=0m−1〈qs,x,qk+m−s〉++〈q0,〈qk∣+m+1,ux〉〉+〈〈q0,qk+m,x〉+∑s=1m〈qs,qk+m−s,x〉︸m−1=〈qm,qk,x〉+∑s=0m−1〈qs,qk+m−s,x〉+∑s=1m〈qs,〈qk+m+1−s,ux〉〉==〈qm,qk,x〉+Υk,m−1,x+∑s=0m〈〈qs,ux〉,qk+m+1−s〉︸=〈〈qs,qk+m+1−s〉,ux〉−〈〈qk+m+1−s,ux〉,qs〉+〈〈qk+m+1−s,ux〉,qs〉==〈Υk,m,ux〉+〈qm,qk,x〉+Υk,m−1,x.

Proof of the equality ϒ_k,m,z = ⟨ϒ_k,m, u_y⟩+ ⟨q_m, q_k,y⟩+ϒ_{k, m}_−1,_y is analogus.

Remark 4.3.

We note that the nonlocal symmetries ϒ_k are similar to the nonlocal symmetries for the four-dimensional Martínez Alonso–Shabat equation found in [29], but the Lie brackets on C^∞(ℝ²) here and in the constructions of [29] are different.

Theorem 4.3.

Eq. (1.1) has ‘invisible’ symmetries (symmetries with the zero shadow) in the covering (4.1) defined as

Γk=(0,…,0︸k,q0,q1,q2,…,qm,…), k≥1.

The proof is similar to the proof of theorem 4.2 and is therefore omitted.

5. The structure of the algebra of nonlocal symmetries

The structure of the algebra s˜ of nonlocal symmetries Φm(A),Ψk,Υi,Γj of Eq. (1.1) in the covering (4.1) is described by the following theorem.

Theorem 5.1.

The Jacobi brackets of lifts of the local symmetries are lifts of the Jacobi brackets of the corresponding local symmetries, that is, the commutators for Φ_m(A), Ψ_k satisfy equations (3.1)—(3.5) with φ_m(A), ψ_k replaced by Φ_m(A), Ψ_k, respectively. The other Jacobi brackets are

{Φm(A),Υk}={Φm(A),Γi}={Υk,Υl}={Γi,Γj}=0, 0≤m≤3, k,l≥0, i,j≥1,{Ψ1,Υk}=(k+3)Υk, {Ψ2,Υk}=−2kΥk, {Ψ3,Υk}=−(k+1)Υk+1,{Ψ1,Γi}=−(i−1)Γi, {Ψ2,Γi}=2(i−1)Γk, {Ψ3,Γi}=(i−1)Γi−1,{Γi,Υk}={−Υk−i+1,k≥i−1,0,k<i−1.

Proof. We will prove that {Φ₀(A), ϒ_k} = 0 . The proof for the other Jacobi brackets is similar and therefore is omitted.

We start with writing down explicitly the formula for computing the Jacobi bracket in case of some infinite-dimensional vectors Θ and Ω. In order to facilitate applying this formula to vectors Φ₀(A) and ϒ_k we enumerate coordinates of the vectors Θ and Ω in the following way: Θ = (Θ₋₁, Θ₀,…, Θ_m,…), Ω = (Ω₋₁, Ω₀, …, Ω_m,…) . Then the Jacobi bracket is of the form {Θ, Ω} = ({Θ, Ω}₋₁,{Θ, Ω}₀,{Θ, Ω}₁,…), where

{Θ,Ω}j=∑I(D˜I(Θ−1)∂∂uI(Ωj)−D˜I(Ω−1)∂∂uI(Θj)+ +∑m=0∞(D˜I(Θm)∂∂qm,I(Ωj)−D˜I(Ωm)∂∂qm,I(Θj))), j≥−1.

As for coordinates q_m_,_I, Φ_0,_j(A) depends at most on q_j_,_t, q_j_,_x, q_j_,_y, q_j_,_z, q_j or q_j _{+ 1} and ϒ_k_,_j depends at most on q_m_,_x, q_k ₊ _j _{+ 1−}_m_,_x, q_m_,_y, q_k ₊ _j _{+ 1−}_m_, _y for m = 0, …, j . The following observations will be used frequently:

∑I∑m=0∞D˜I(am)∂∂qm,I(Φ0,j)=∑l=t,x,y,zD˜l(aj)∂∂qj,l(Φ0,j), j≥−1,i=0,1,2,3.∑I∑m=0∞D˜I(am)∂∂qm,I(Υk,j)=∑m=0j(D˜x(am)qk+j+1−m,y−D˜x(ak+j+1−m)qm,y−D˜y(am)qk+j+1−m,x+D˜y(ak+j+1−m)qm,x)=∑m=0j〈am,qk+j+1−m〉−〈ak+j+1−m,qm〉.

We have

{Φi(A),Υk}−1=∑ID˜I(ϕi(A))∂∂uI(qk)︸=0−∑ID˜I(qk)∂∂uI(φi(A))++∑I∑m=0∞D˜I(Φi,m(A))∂∂qm,I(qk)︸=Φi,k(A)−∑I∑m=0∞D˜ I(Υk,m)∂∂qm,I(φi(A))︸=0, 0≤i≤3.

It is easy to see, that ∑ID˜I(qk)∂∂uI(φi(A))=Φik(A) for i = 0,1,2,3, hence {Φ_i(A), ϒ_k}₋₁ = 0 . Then for j ≥ 0 we get

{Φi(A),Υk}j=∑ID˜I(φi(A))∂∂uI(Υk,j)︸=0−∑ID˜I(qk)∂∂uI(Φi,j(A))︸=0++∑I∑m=0∞(D˜I(Φi,m(A))∂∂qm,I(Υk,j))−∑I∑m=0∞(D˜I(Υk,m)∂∂qm,I(Φi,j(A))).

Furthermore, we have

{Φi(A),Υk}j==∑l=x,y∑m=0∞(D˜l(Φ0,m(A))∂∂qm,l(Υk,j))−∑l=t,x,y,z∑m=0∞(D˜l(Υk,m)∂∂qm,l(Φ0,j(A)))==∑m=0j(D˜x(Φ0,m(A))qk+j+1−m,y−D˜x(Φ0,k+j+1−m(A))qm,y−D˜y(Φ0,m(A))qk+j+1−m,x++D˜ y(Φ0,k+j+1−m(A))qm,x)−∑l=t,x,y,zD˜l(Υk,j)∂∂qj,l(Φ0,j(A))==∑m=0J(〈Φ0,m(A),qk+j+1−m〉−〈Φ0,k+j+1−m(A),qm〉)−−∑l=t,x,y,zD˜l(∑m=0j〈qm,qk+j+1−m〉)∂∂qj,l(Φ0,j(A))==∑m=0j(〈Φ0,m(A),qk+j+1−m〉−〈Φ0,k+j+1−m(A),qm〉=−∑l=t,x,y,z(〈qm,l,qk+j+1−m〉+〈qm,qk+j+1−m,l〉)∂∂qj,l(Φ0,j(A))==∑m=0i〈Φ0,m(A)∣,qk+j+1−m〉−〈Φ0,k+j+1−m(A),qm〉++〈Φ0,k+j+1−m(A),qm〉−〈Φ0m(A),qk+j+1−m〉)=0

Remark 5.1.

Denote by 𝔞 the ideal of the Lie algebra s˜ generated by ϒ_i and by 𝔟 the subalgebra generated by Γ_j. Identify 𝔰_∞ and s⋄ with the ideal generated by Φ_m(A) and the subalgebra generated by Ψ_k, respectively. Then we obtain

s˜=(s∞⊕(a×b))×s⋄

(here ⊕ denotes a direct sum of commuting Lie algebras), in particular we have equations (1.3) and (1.4) together with[s∞,s∞]=s∞[s⋄,s∞]=s∞, and [s⋄,s⋄]=s⋄.

6. Conclusion

In this paper we have found two infinite hierarchies of commuting nonlocal symmetries for Plebański’s second heavenly equation. One of these hierarchies can be employed to construct new solutions using the techniques presented e.g. in [10,19,31], see also [5,16,17,34]. Furthermore, we find the lifts of local symmetries and the structure of the Lie algebra of the nonlocal symmetries. We emphasize that finding an explicit form of nonlocal symmetries and the commutator relations for an infinite-dimensional symmetry algebra of nonlocal symmetries (rather than just shadows) is quite rare. We know just a number of similar results in the literature, [1–3,5,13,16–18,29]. We hope that it would be very interesting to study how the infinite-dimensional Lie algebra of nonlocal symmetries reflects the algebraic structure behind the integrability properties of the considered equation, cf. [27,28].

Acknowledgments

The authors gratefully acknowledge financial support from the Polish Ministry of Science and Higher Education. We are pleased to thank A. Sergyeyev for important and stimulating discussions, to E.G. Reyes for pointing out the references [5,16,17], and to the anonymous referee for useful suggestions.

Footnotes

a

We identify here the Lie algebra of local symmetries with its lift to nonlocal symmetries in the covering (4.1)

b

See definition of an Abelian covering in Section 2

c

We carried out all computations in the Jets software [4].

d

see remark 4.1.

References

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Journal: Journal of Nonlinear Mathematical Physics
Volume-Issue: 25 - 2
Pages: 188 - 197
Publication Date: 2021/01/06
ISSN (Online): 1776-0852
ISSN (Print): 1402-9251
DOI: 10.1080/14029251.2018.1452669 How to use a DOI?
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TY  - JOUR
AU  - Aleksandra Lelito
AU  - Oleg I. Morozov
PY  - 2021
DA  - 2021/01/06
TI  - Nonlocal symmetries of Plebański’s second heavenly equation
JO  - Journal of Nonlinear Mathematical Physics
SP  - 188
EP  - 197
VL  - 25
IS  - 2
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2018.1452669
DO  - 10.1080/14029251.2018.1452669
ID  - Lelito2021
ER  -

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