Journal of Nonlinear Mathematical Physics

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

Nonlocal symmetries of Plebański’s second heavenly equation

Aleksandra Lelito, Oleg I. Morozov
Faculty of Applied Mathematics, AGH University of Science and Technology, Al. Mickiewicza 30, Cracow 30-059, Poland,,
Received 20 July 2017, Accepted 7 November 2017, Available Online 6 January 2021.
10.1080/14029251.2018.1452669How to use a DOI?
Plebański’s second heavenly equation; Lax pair; differential covering; nonlocal symmetries

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.

© 2018 The Authors. Published by Atlantis and Taylor & Francis
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1. Introduction

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

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,
with a non-removable parameter λ. Expanding the pseudopotential q into a Taylor series q=k=0λkqk yields a new covering (4.1) with pseudopotentials qk 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=ss, 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

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 algebraa 𝔰 ⊕ 𝔞 ⊕ 𝔟 (the sum as vector spaces). In paricular, we show that

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:n×mn,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,…,in) is a multi-index, and for every local section f:nn×m of π the corresponding infinite jet j(f) is a section j(f):nJ(π) such that uIα(j(f))=#IfαxI=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=uttxxyyzz with i times t, j times x, k times y, and l times z

The vector fields

(i1,…,ik,…,in) + 1k = (i1,…,ik + 1, …,in), 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)m is the vector field

with DI=D(i1,in)=Dx1i1Dxnin.

A system of PDEs Fr(xi,uIα)=0,#Is,r{1,,R}, of the order s ≥ 1 with R ≥ 1 defines the submanifold ={(xi,uIα)J(π)DK(Fr(xi,uIα))=0,#K0} in J(π).

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

where ε=Fε with the matrix differential operator
Solutions to (2.2) constitute the Lie algebra Lie() of (infinitesimal) symmetries of equation with respect to the Jacobi bracket { φ, ψ} = Eφ(ψ) − Eψ(φ). The subalgebra of contact symmetries of is Lie0()=Lie()C(J1(π),m).

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

such that [D˜xi,D˜xj]=0 for all ij whenever (xi,uIα). We define the partial derivatives of ws by wxks=D˜xk(ws). This yields the system of covering equations
This over-determined system of PDEs is compatible whenever (xi,uIα).

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

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 corresponding to the covering τ with the extended total derivatives (2.3), or τ-shadow, is a function φC(×W,m), such that

is a consequence of equations DK(F) = 0 and (2.4). A nonlocal symmetry of corresponding to the covering τ (or τ -symmetry) is the vector field
with AsC(×W) such that φ satisfies (2.5) and
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 As in (2.6). But for any τ-shadow φ there exists a covering τφ and a nonlocal τφ-symmetry whose τφ-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, ut, ux, uy, uz) to the equation

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

Remark 3.1.

We have s=ss, where s is generated by φi(A) and s is generated by ψj 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 [fV,gW ] = f g ⊗ [V,W ] for f ,g ∈ ℝ4[s] and V, W ∈ 𝔥.

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


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

Direct computations prove:

Proposition 4.1.

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

Then we have:

Corollary 4.1.

Functions υk = qk, k ≥ 0, are shadows in the covering (4.1).

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

and the linearization
of the system (4.1).

Remark 4.1.

To simplify notation, here and below we use φxz 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 Φi(A) = (φi(A), Φi,0(A), Φi,1(A), …, Φi,k(A), …), Ψj = (ψj, Ψj,0, Ψj,1, …, Ψj,k, …), with


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

Theorem 4.2.

The shadows vk = qk, k ≥ 0, have the lifts ϒk to the nonlocal symmetries in the covering (4.1) defined as ϒk = (qk, ϒk,0, ϒk,1, … , ϒk,m, …) with

wherea, b⟩ = axbyay bx

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 thatd

Proof of the equality ϒk,m,z = ⟨ϒk,m, uy⟩+ ⟨qm, qk,y⟩+ϒk, m−1,y is analogus.

Remark 4.3.

We note that the nonlocal symmetries ϒk 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


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 Φm(A), Ψk satisfy equations (3.1)(3.5) with φm(A), ψk replaced by Φm(A), Ψk, respectively. The other Jacobi brackets are


Proof. We will prove that {Φ0(A), ϒk} = 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 ϒk we enumerate coordinates of the vectors Θ and Ω in the following way: Θ = (Θ−1, Θ0,…, Θm,…), Ω = (Ω−1, Ω0, …, Ωm,…) . 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(Θ1)uI(Ωj)D˜I(Ω1)uI(Θj)+          +m=0(D˜I(Θm)qm,I(Ωj)D˜I(Ωm)qm,I(Θj))),j1.
As for coordinates qm,I, Φ0,j(A) depends at most on qj,t, qj,x, qj,y, qj,z, qj or qj + 1 and ϒk,j depends at most on qm,x, qk + j + 1−m,x, qm,y, qk + j + 1−m, y for m = 0, …, j . The following observations will be used frequently:
We have
{Φi(A),Υk}1=ID˜I(ϕi(A))uI(qk)=0ID˜I(qk)uI(φi(A))++Im=0D˜I(Φi,m(A))qm,I(qk)=Φi,k(A)Im=0D˜ I(Υk,m)qm,I(φi(A))=0,0i3.
It is easy to see, that ID˜I(qk)uI(φi(A))=Φik(A) for i = 0,1,2,3, hence {Φi(A), ϒk}−1 = 0 . Then for j ≥ 0 we get
Furthermore, we have
{Φi(A),Υk}j==l=x,ym=0(D˜l(Φ0,m(A))qm,l(Υk,j))l=t,x,y,zm=0(D˜l(Υk,m)qm,l(Φ0,j(A)))==m=0j(D˜x(Φ0,m(A))qk+j+1m,yD˜x(Φ0,k+j+1m(A))qm,yD˜y(Φ0,m(A))qk+j+1m,x++D˜ y(Φ0,k+j+1m(A))qm,x)l=t,x,y,zD˜l(Υk,j)qj,l(Φ0,j(A))==m=0J(Φ0,m(A),qk+j+1mΦ0,k+j+1m(A),qm)l=t,x,y,zD˜l(m=0jqm,qk+j+1m)qj,l(Φ0,j(A))==m=0j(Φ0,m(A),qk+j+1mΦ0,k+j+1m(A),qm=l=t,x,y,z(qm,l,qk+j+1m+qm,qk+j+1m,l)qj,l(Φ0,j(A))==m=0iΦ0,m(A),qk+j+1mΦ0,k+j+1m(A),qm++Φ0,k+j+1m(A),qmΦ0m(A),qk+j+1m)=0

Remark 5.1.

Denote by 𝔞 the ideal of the Lie algebra s˜ generated by ϒi and by 𝔟 the subalgebra generated by Γj. Identify 𝔰 and s with the ideal generated by Φm(A) and the subalgebra generated by Ψk, respectively. Then we obtain

(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, [13,5,13,1618,29]. 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].


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.



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


See definition of an Abelian covering in Section 2


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


see remark 4.1.


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Journal of Nonlinear Mathematical Physics
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Cite this article

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  -
DO  - 10.1080/14029251.2018.1452669
ID  - Lelito2021
ER  -