---
abstract: 'We consider the 3D equation $u_{yy} = u_{tx} + u_yu_{xx} - u_xu_{xy}$ and its 2D reductions: (1) $u_{yy} = (u_y+y)u_{xx}-u_xu_{xy}-2$ (which is equivalent to the Gibbons-Tsarev equation) and (2) $u_{yy} = (u_y+2x)u_{xx} + (y-u_x)u_{xy} -u_x$. Using reduction of the known Lax pair for the 3D equation, we describe nonlocal symmetries of (1) and (2) and show that the Lie algebras of these symmetries are isomorphic to the Witt algebra.'
address:
- 'Mathematical Institute, Silesian University in Opava, Na Rybníčku 1, 746 01 Opava, Czech Republic'
- 'Independent University of Moscow, B. Vlasevsky 11, 119002 Moscow, Russia & Trapeznikov Institute of Control Sciences, 65 Profsoyuznaya street, Moscow 117997, Russia'
- 'Faculty of Applied Mathematics, AGH University of Science and Technology, Al. Mickiewicza 30, Kraków 30-059, Poland'
- 'Mathematical Institute, Silesian University in Opava, Na Rybníčku 1, 746 01 Opava, Czech Republic'
author:
- 'P. Holba'
- 'I.S. Krasil[[[$''$]{}]{}]{}shchik'
- 'O.I. Morozov'
- 'P. Voj[č]{}[á]{}k'
date:
- 
- 
title: |
    2D reductions of the equation $\mathbf{u_{yy} = u_{tx} + u_yu_{xx} - u_xu_{xy}}$\
    and their nonlocal symmetries
---

Introduction {#introduction .unnumbered}
============

The equation $$\label{eq:1}
  u_{yy} = u_{tx} + u_yu_{xx} - u_xu_{xy}$$ belongs to the class of linearly degenerate integrable equations [@Fer-Moss-2015] and was, as far as we know, introduced first in [@Mikhalev-1992] and independently in [@Pavlov-2003; @Dunajski-2004]. In [@B-K-M-V-2014], we described all two-dimensional symmetry reductions of this equation. All these reductions are either linearizable or exactly solvable, except for the two ones: $$\label{eq:2}
  u_{yy} = (u_y+y)u_{xx}-u_xu_{xy}-2$$ and $$\label{eq:3}
  u_{yy} = (u_y+2x)u_{xx} + (y-u_x)u_{xy} -u_x.$$ Equation  is reduced to the Gibbons-Tsarev equation [@Gibbons-Tsarev-1996] by the simple transformation $u \mapsto u-y^2/2$.

Equation  admits the Lax pair $$\label{eq:4}
  \begin{array}{lcl}
    w_{t}&=&(\lambda^2-\lambda u_x-u_y)w_x,\\
    w_{y}&=&(\lambda-u_x)w_{x}
  \end{array}$$ with the non-removable spectral parameter $\lambda$. In [@B-K-M-V-2015], we, in particular, studied the behavior of this Lax pair under the symmetry reduction.

In this paper, we use system  to describe nonlocal symmetries of Equations  and  and prove that in both cases these symmetries form the Lie algebra isomorphic to the Witt algebra $$\mathfrak{W} = {\{\,e_i=z^{i+1}\textstyle{\frac{\partial}{\partial z}} \midi\in \mathbb{Z}\,\}}.$$ In Section \[sec:preliminaries\], we recall some necessary results obtained in the previous research and introduce the notions and constructions needed for the subsequent exposition. Section \[sec:main-result\] is devoted to the proofs of the basic results. We discuss in detail Equation  and briefly repeat the main step for the second equation, because the reasoning is quite similar in both cases. The obtained results are discussed in Section \[sec:discussion\] together with further perspectives.

Preliminaries {#sec:preliminaries}
=============

We recall here basic facts from the theory of nonlocal symmetries (see [@Kras-Vin-Trends-1989]) and previous results on Equation  and its reductions, see [@B-K-M-V-2014; @B-K-M-V-2015].

Basics {#sec:basics}
------

Consider a differential equation[^1] $$\label{eq:5}
  F^\alpha\left(x,\dots,\frac{\partial^{{\left|\sigma\right|}}u}{\partial
      x^\sigma},\dots\right) = 0,\qquad \alpha=1,\dots,r,$$ of order $k$ in unknowns $u=(u^1,\dots,u^m)$, where $u^j=u^j(x)$, $x=(x^1,\dots,x^n)$. To the infinite prolongation of  there corresponds a locus $\mathcal{E}\subset J^\infty(\pi)$ in the space of infinite jets of the trivial bundle $\pi\colon \mathbb{R}^m\times\mathbb{R}^n \to \mathbb{R}^n$, see [@AMS-book]. Consider jet coordinates $u_\sigma^j$ on $J^\infty(\pi)$, $j=1,\dots,m$, where $\sigma=i_1\dots i_s$ is a symmetric multi-index, $i_\alpha=1,\dots,n$, and choose a set ${\mathbf{I}}=\{u_\sigma^j\}$ of internal variables on $\mathcal{E}$. Denote by $$D_{x^i} = \frac{\partial}{\partial x^i} + \sum_{{\mathbf{I}}}u_{\sigma
    i}\frac{\partial}{\partial u_\sigma^j},\qquad i=1,\dots,n,$$ the operators of total derivatives on $\mathcal{E}$. For any function $f$ on $J^\infty(\pi)$ we denote by $\ell_f$ the restriction of its linearization $\sum_{j,\sigma} \partial f/\partial u_\sigma^j D_\sigma$ to $\mathcal{E}$, where $D_\sigma = D_{x^{i_1}}\circ\dots\circ D_{x^{i_s}}$. A (local) symmetry of $\mathcal{E}$ is an evolutionary vector field $${\mathbf{E}}_\phi = \sum_{{\mathbf{I}}}D_\sigma(\phi^j)\frac{\partial}{\partial u_\sigma^j},$$ where the generating function $\phi=(\phi^1,\dots,\phi^m)$ satisfies the equation $\ell_F(\phi)=0$ and $F=(F^1,\dots,F^r)$ determines Equation . We identify symmetries with their generating functions and denote the Lie algebra of symmetries by $\operatorname{sym}\mathcal{E}$.

A (differential) covering over $\mathcal{E}$ is a bundle $\tau\colon \tilde{\mathcal{E}}= \mathcal{E}\times \mathbb{R}^l\to
\mathcal{E}$, $l=1,2,\dots,\infty$, endowed with vector fields $$\tilde{D}_{x^i} = D_{x^i} + \sum_\alpha X_i^\alpha\frac{\partial}{\partial
    w^\alpha},\qquad i=1,\dots,n,$$ that pair-wise commute, where $w^\alpha$ are coordinates in $\mathbb{R}^l$ called nonlocal variables. Equivalently, $\tilde{\mathcal{E}}$ may be regarded as an overdetermined system $$w_{x^i}^\alpha = X_i^\alpha,\qquad i=1,\dots,n,\quad \alpha = 1,\dots,l,$$ whose compatibility conditions are consequences of $\mathcal{E}$. For a linear differential operator $\Delta = \left(\sum_\sigma a_{ij}^\sigma D_\sigma\right)$ on $\mathcal{E}$ we denote by $\tilde{\Delta} = \left(\sum_\sigma a_{ij}^\sigma \tilde{D}_\sigma\right)$ its lift to $\tilde{\mathcal{E}}$.

Coverings $\tau_1$ and $\tau_2$ are said to be equivalent if there exists a diffeomorphism $g\colon \tilde{\mathcal{E}}_1 \to \tilde{\mathcal{E}}_2$ such that $\tau_2\circ g = \tau_1$ and which takes $\tilde{\mathcal{C}}_1$ to $\tilde{\mathcal{C}}_2$, where $\tilde{\mathcal{C}}_i$ is the span of the fields $\tilde{D}_{x^i}$ on $\tilde{\mathcal{E}}_i$.

A nonlocal symmetry of $\mathcal{E}$ in the covering $\tau$ is a vector field $${\mathbf{S}}_\Phi = \sum_{\mathbf{I}}\tilde{D}_\sigma(\phi^j)\frac{\partial}{\partial
    u_\sigma^j} +
  \sum_\alpha \psi^\alpha\frac{\partial}{\partial w^\alpha},$$ where $\phi=(\phi^1,\dots,\phi^m)$ and $\psi^\alpha$ are functions on $\tilde{\mathcal{E}}$ satisfying the system $$\begin{aligned}
  &\tilde{\ell}_F(\phi)=0,\\
  &\tilde{D}_{x^i}(\psi^\alpha) = \tilde{\ell}_{X_i^\alpha}(\phi) + \sum_\beta
    \frac{\partial X_i^\alpha}{\partial w^\beta}\end{aligned}$$ for all $i$ and $\alpha$. Any nonlocal symmetry is identified with the collection $\Phi = (\phi,\psi^1,\dots,\psi^\alpha,\dots)$. A symmetry is called a lift of a local symmetry if the component $\phi$ is a function on $\mathcal{E}$; it is called invisible if $\phi=0$. Nonlocal symmetries form a Lie algebra denoted by $\operatorname{sym}_\tau\mathcal{E}$.

Coverings over Equation  and symmetries {#sec:cover-over-equat}
---------------------------------------

Let us assume that $w=w(\lambda)$ in Equation  and consider the expansion $$w=\sum_{i=-\infty}^{+\infty}\lambda^{-i} w_i.$$ This leads to the infinite-dimensional covering $$\label{eq:6}
  \begin{array}{rcl}
    w_{i,x} &=& w_{i-1,y} + u_x w_{i-1,x} ,\\
    w_{i,y} &=& w_{i-1,t} + u_y w_{i-1,x},
  \end{array}$$ where $i\in\mathbb{Z}$.

Recall now that the space $\operatorname{sym}\mathcal{E}$ for Equation  is spanned by the functions $$\begin{gathered}
   \theta_1=2x-yu_x,\qquad   \theta_2=3u-2xu_x-yu_y,\\
   \theta_3(T)=Tu_y+T'(yu_x-x)-\frac{1}{2}T''y^2, \qquad   \theta_4(T)=Tu_x-T'y,\\
   \theta_5(T)=Tu_t+T'(xu_x+yu_y-u)+\frac{1}{2}T''(y^2u_x-2xy)-
  \frac{1}{6}T'''y^3,  \qquad
  \theta_6(T)=T,\end{gathered}$$ where $T$ is a function in $t$ and ‘prime’ denotes the $t$-derivatives. In what follows, we shall need the following

\[prop:cover-over-equat-1\] The symmetries $\theta_1$ $\theta_2$ and $\theta_5 = \theta_5(1)= u_t$ can be lifted to the covering .

Let us set $$\begin{aligned}
    \theta_1^i&= -y w_{i,x} + (i + 2) w_{i-1},\\
    \theta_2^i&= -2 x w_{i,x} - y w_{i,y} + (i + 3) w_i,\\
    \theta_5^i&= w_{i,t}.
  \end{aligned}$$ Then $$\Theta_j=(\theta_j,\dots,\theta_j^i,\dots),\qquad j=1,2,5,\quad
    i\in\mathbb{Z},$$ are the desired lifts.

Reductions {#sec:reductions}
----------

Using Proposition \[prop:cover-over-equat-1\], we now state the following

\[prop:reductions-2\] Reduction of  with respect to the symmetry $\Theta_5+\Theta_1$ leads to the covering $$\label{eq:7}
    \begin{array}{rcl}
      w_{i,x} &=& w_{i-1,y} + u_x w_{i-1,x} ,\\
      w_{i,y} &=& (u_y + y)w_{i-1,x} - (i + 1)w_{i-2}
    \end{array}$$ over Equation  while reduction with respect to $\Theta_5 + \Theta_2$ leads to the covering $$\label{eq:8}
    \begin{array}{rcl}
      w_{i,x} &=& w_{i-1,y} + u_x w_{i-1,x} ,\\
      w_{i,y} &=& (u_y + 2x)w_{i-1,x} + yw_{i-1,y} - (i + 2)w_{i-1}
    \end{array}$$ over Equation .

Let us finally describe the algebras of local symmetries for Equations  and . Direct computations show that $\operatorname{sym}\mathcal{E}_1$ for the first equation is spanned by the functions $$\label{eq:12}
  \phi_{-4} = 1,\quad \phi_{-3} = u_x,\quad \phi_{-2}= -u_y - y,\quad
  \phi_{-1} = -2 x + yu_x,\quad
  \phi_0 = 4 u - 3 xu_x - 2 yu_y,$$ while in the second case we have the following generators of the symmetry algebra $\operatorname{sym}\mathcal{E}_2$: $$\gamma_{-3} = 1,\quad \gamma_{-2}=-y-\frac{1}{2}u_x,\quad \gamma_{-1} =  y^2  - 2
  x + 2 yu_x - 2 u_y,\quad \gamma_0= 3 u - 2x u_x - yu_y.$$ It is also easily seen that $$\left.\begin{array}{l}
    \text{for Equation~\eqref{eq:2}}\\
     \protect{[\phi_i,\phi_j]= }
    \begin{cases}
      (j-i)\phi_{i+j}&\text{if } i+j\geq -4,\\
      0&\text{otherwise},
    \end{cases}
  \end{array}\qquad\right|\qquad
  \begin{array}{l}
    \text{for Equation~\eqref{eq:3}}\\
    \protect{[\gamma_i,\gamma_j]= }
    \begin{cases}
      (j-i)\gamma_{i+j}&\text{if } i+j\geq -3,\\
      0&\text{otherwise}.
    \end{cases}
  \end{array}$$ Denote by $\mathfrak{W}_0\subset\mathfrak{W}$ the subalgebra in the Witt algebra spanned by the fields $e_i$ with $i\leq 0$ and by $\mathfrak{W}_k$ its ideal spanned by $e_i$, $i\leq k < 0$.

\[prop:reductions-1\] For Equation  one has $\operatorname{sym}\mathcal{E}_1=\mathfrak{W}_0/\mathfrak{W}_5$ while $\operatorname{sym}\mathcal{E}_2=\mathfrak{W}_0/\mathfrak{W}_4$ for Equation .

The coverings $\tau^p$ and $\sigma^p$ {#sec:coverings-taup}
-------------------------------------

Consider the covering  over Equation . Choose an integer $p\in\mathbb{Z}$ and assume that $w_i=0$ for all $i<p$. To proceed further, it is convenient to relabel nonlocal variables by setting $w_{p+i}=r_{i-4}^p$. Then $r_i^p=0$ for $i<-4$ and $$\begin{aligned}
  &r_{-4,x}^p=0,&&r_{-4,y}^p=0;\\
  &r_{-3,x}^p=0,&&r_{-3,y}^p=0,\end{aligned}$$ and without loss of generality one can set $r_{-4}^p=1$, $r_{-3}^p=0$. Then $$r_{-2}^p=-(p + 3)y,\qquad r_{-1}^p = -(p + 3)x,\qquad r_0^p = -(p + 3)u +
  \frac{1}{2}(p + 3)(p + 4)y^2,$$ while $$\label{eq:9}
  \begin{array}{rcl}
    r_{i,x}^p&=&r_{i-1,y}^p + u_xr_{i-1,x}^p,\\[3pt]
    r_{i,y}^p&=&(u_y+y)r_{i-1,x}^p-(p+i+5)r_{i-2}^p,
  \end{array}$$ for all $i\geq 1$. We denote this covering by $\tau^p$.

In a similar way, for Equation  and its covering  we have $r_i^p=0$ for $i<-3$, $r_{-3}^p = 1$, and $$r_{-2}^p = -(p + 3)y,\quad r_{-1}^p = -(p + 3)x +
  \frac{1}{2}(p + 3)^2y^2,\quad r_0^p = - (p + 3)u + (p + 3)^2 xy -
  \frac{1}{6} (p + 3)^3 y^3,$$ while $$\label{eq:10}
  \begin{array}{rcl}
    r_{i,x}^p&=& r_{i-1,y}^p + u_x r_{i-1,x}\\
    r_{i,y}^p&=& (u_y + 2x)r_{i-1,x}^p + yr_{i-1,y}^p - (p + i + 5)r_{i-1}^p
  \end{array}$$ for $i\geq1$. This covering will be denoted by $\sigma^p$.

The main result {#sec:main-result}
===============

We prove the main result of the paper in this section, which states that in coverings $\tau^p$, naturally associated with  and  the algebras of nonlocal symmetries for both Equations  and  are isomorphic to the Witt algebra $\mathfrak{W}$. The case of Equation  is considered in detail, while for Equation  we provide a sketch of proofs only.

Equivalence of the coverings $\tau^p$ {#sec:equiv-cover-taup}
-------------------------------------

The first step of the proof is to establish the equivalence of different $\tau^p$ to each other.

\[prop:equiv-cover-taup-1\] The covering $\tau^p$ is equivalent to $\tau^{-4}$ for any $p$.

We distinguish the two cases: $p\neq -3$ and $p=3$.

**The case $p\neq -3$.** Introduce the notation $s_i=r_i^{-4}$ and consider the vector field $$\mathcal{X}=x\frac{\partial}{\partial y} + 2u\frac{\partial}{\partial x} +
    3s_1\frac{\partial}{\partial u} + \sum_{i\geq
      1}(i+3)s_{i+1}\frac{\partial}{\partial s_i}.$$ Let us define the quantities $Q_{k,j}$, $k\geq0$, by $$Q_{k,0} = \frac{1}{(k+2)!}y^{k+2},\qquad Q_{k,j+1} =
    \frac{1}{j}\mathcal{X}(Q_{k,j})$$ and assume $Q_{k,j}=0$ for $j<0$. Set $$d_i = \sum_{k\geq 0}(-1)^k\frac{(p+k+4)!}{(p+4)!}Q_{k,i-2k}.$$ Then the transformation $$r_i^p = -(p+3)(s_i-(p+4)d_i),\qquad i\geq1,$$ is the desired equivalence.

**The case $p= -3$.** The first three pairs of defining equations in this case are $$r_{i,x}=0,\quad r_{i,y}=0,\qquad i=-3,-2,-1,$$ while all the rest ones do not contain the variable $r_{-3}$. Thus we that $r_{-3}$ is a constant whose value does not influence the subsequent computations. So, we may set $r_{-3}=0$ and this reduces the case under consideration to $p>-3$.

The Lie algebra of nonlocal symmetries {#sec:lie-algebra-tructure}
--------------------------------------

Let $\Phi=(\phi,\phi^1,\dots,\phi^i,\dots)$ be a nonlocal symmetry of Equation  in the covering . The defining equations for the components of $\Phi$ are $$\begin{aligned}
\nonumber
  \tilde{\ell}_F(\phi)&\equiv \tilde{D}_y^2(\phi) - (u_y +
  y)\tilde{D}_x^2(\phi) - u_{xx}\tilde{D}_y(\phi)  + u_x
  \tilde{D}_x\tilde{D}_y(\phi)  + u_{xy}\tilde{D}_x(\phi)=0, \\\label{eq:11}
  \tilde{D}_x(\phi^i)&=\tilde{D}_y(\phi^{i-1}) + u_x\tilde{D}_x(\phi^{i-1}) +
  r_{i-1,x}^p\tilde{D}_x(\phi), \\ \nonumber
  \tilde{D}_y(\phi^i)&=(u_y + y)\tilde{D}_x(\phi^{i-1}) - (p + i +
  1)\phi^{i-2} + r_{i-1,x}^p\tilde{D}_y(\phi).\end{aligned}$$ We need a number of auxiliary results to describe the algebra $\operatorname{sym}_{\tau^p}\mathcal{E}_1$. As a first step, it is convenient to assign weights to the internal coordinates in $\tilde{\mathcal{E}}_1$ in such a way that all polynomial objects become homogeneous with respect to these weights. Let us set $${\left|x\right|} = 3,\qquad {\left|y\right|} = 2,\qquad {\left|u\right|} = 4.$$ Then $${\left|u_x\right|} = {\left|u\right|} - {\left|x\right|} = 1,\qquad {\left|u_y\right|} = {\left|u\right|} - {\left|y\right|} = 2,$$ etc., and $${\left|r_i^p\right|} = i+4,\qquad i\geq1.$$ The weight of a monomial is the sum of weights of its factors and the weight of a vector field $R\partial/\partial \rho$ is ${\left|R\right|}-{\left|\rho\right|}$. In particular, one has ${\left|\phi_k\right|}=k$, $k=-4,\dots,0$ for the local symmetries presented in .

\[lem:lie-algebra-nonlocal\] The local symmetry $\phi_{-1}$ can be lifted to the covering $\tau^{-4}$.

Let us set $\Phi_{-1}=(\phi_{-1},\phi_{-1}^1,\dots,\phi_{-1}^i,\dots)$, where $\phi_{-1}^i=yr_{i,x}^{-4}-(i+2)r_{i-1}^{-4}$. It is straightforward to check that Equations  are satisfied for $p=-4$.

\[lem:lie-algebra-nonlocal-1\] The local symmetry $\phi_{-2}$ can be lifted to the covering $\tau^{-5}$.

We set $\Phi_{-2} =
  (\phi_{-2},-r_{1,y}^{-5},\dots,-r_{i,y}^{-5},\dots)$. Then  are fulfilled in an obvious way for $p=-5$.

\[lem:lie-algebra-nonlocal-2\] There exists a nonlocal symmetry of weight $2$ in the covering $\tau^{-1}$.

Let us set $\Phi_2=(\phi_2,\phi_2^1,\dots,\phi_2^i,\dots)$, where $$\phi_2=
    -\frac{1}{2}\,\left(8 r_2^{-1}+42 y (3 y^2-2 u)+(80 x y-7 r_1^{-1}) u_x
    +2 (7 y^2+6 u) u_y-50   x^2-96 y^3\right)$$ and $$\phi_2^i=
    -\frac{1}{2}\,\left(
    (80 x y-7 r_1^{-1}) r_{i, x}^{-1}+2 (7 y^2+6 u) r_{i, y}^{-1}
    -2 (i+8) r_{i+2}^{-1}-8 y (i+6) r_{i}^{-1}-10 (i+5) x r_{i-1}^{-1}
    \right)$$ for all $i\geq1$.

\[prop:lie-algebra-nonlocal\] The symmetries $\Phi_{-2}$ $\Phi_{-1}$ and $\Phi_2$ exist in any covering $\tau^p$.

A direct consequence of Lemmas \[lem:lie-algebra-nonlocal\]– \[lem:lie-algebra-nonlocal-2\] and Proposition \[prop:equiv-cover-taup-1\].

\[thm:lie-algebra-nonlocal\] The symmetries $\Phi_{-2}$ $\Phi_{-1}$ and $\Phi_{2}$ generate the entire Lie algebra $\operatorname{sym}_{\tau^p}\mathcal{E}_1$ which is isomorphic to the Witt algebra $\mathfrak{W}$ i.e. there exists a basis $\Phi_k$ $k\in\mathbb{Z}$ such that $$[\Phi_k,\Phi_l] =(l-k)\Phi_{k+l}$$ for all $k$ $l\in\mathbb{Z}$.

Let set $$\Phi_0=4[\Phi_{-2},\Phi_2],\qquad \Phi_1=3[\Phi_{-1},\Phi_2]$$ and by induction $$\Phi_{k+1} = (k-1)[\Phi_1,\Phi_k],\qquad \Phi_{-k-1} =
    (1-k)[\Phi_{-1},\Phi_{-k}]$$ for all $k\geq 2$.

The symmetries $\Phi_k$ are invisible for $k<-4$.

The symmetries $\Phi_{-4}$, $\Phi_{-3}$, and $\Phi_0$ are lifts of local symmetries $\phi_{-4}$, $\phi_{-3}$, $\phi_0$, respectively.

Explicit formulas {#sec:explicit-formulas}
-----------------

To obtain explicit formulas for nonlocal symmetries in $\tau^p$ for arbitrary[^2] value of $p$ consider the fields $$\mathcal{Y}_m = \sum_{i=m}^\infty(i-m+1)r_{i-3}^p\frac{\partial}{\partial
    r_{i-4}^p}$$ on the space of $\tau^p$, where $m=1\dots,4$, and the quantities $P_{i,j}^m$ defined by induction as follows: $$P_{i,0}^{(m)} = \frac{1}{(i+2)!}\left(r_{m-4}^p\right)^{i+2},\qquad
  P_{i,j+1}^{(m)} = \frac{1}{j+1}\mathcal{Y}_m(P_{i,j}^{(m)}),$$ where $i$, $j\geq 0$, $m=1,\dots,4$. We set $P_{i,j}^{(m)}=0$ if at least one of the subscripts is $<0$. In terms of these quantities, the lift of $\phi_{-2}$ to $\tau^p$ acquires the form $$\Phi_{-2}^p = (\phi_{-2},\phi_{-2}^{1,p},\dots,\phi_{-2}^{i,p},\dots),$$ where $$\phi_{-2}^{i,p} = -r_{i,y}^p - (p+5)\left(r_{i-2}^p + \sum_{j=1}^\infty
    \left(-\frac{1}{p+3}\right)^j\cdot\prod_{l=0}^{j-1}((p+3)l-2)\cdot
    P_{j-1,i-2j}^{(2)}\right).$$ The lift $$\Phi_{-1}^p = (\phi_{-1},\phi_{-1}^{1,p},\dots,\phi_{-1}^{i,p},\dots)$$ of $\phi_{-1}$ is given by $$\phi_{-1}^{i,p} = yr_{i,x}^p - (i+2)r_{i-1}^p + (p+4)\sum_{j=1}^\infty
  \left(-\frac{1}{p+3}\right)^j\cdot  \prod_{l=0}^{j-1}((p+3)l-1)\cdot
  P_{j-1,i-2j+1}^{(2)}.$$ The nonlocal symmetry $\Phi_2$, when passing to $\tau^p$, acquires the form $$\Phi_{2}^p = (\phi_{2}^p,\phi_{2}^{1,p},\dots,\phi_{2}^{i,p},\dots)$$ with $$\begin{aligned}
  \phi_{2}^p&=
  -\frac{1}{p+3}\,\left(
  8r_2^p + 2y(4p+25)r_0^p
      + \Big((p+3)(7p+47)xy - 7r_1^p\Big)u_x\right)
  \\
  &
  -\Big((7y^2+6y)u_y - (4p+29)x^2 - \frac{1}{3}y^3(8p^2 + 87p
  +223)\Big)
  \end{aligned}$$ and $$\begin{aligned}
  \phi_{2}^{i,p}&=
  -\frac{1}{p+3}\,\Big((p+3)(7p+47)xy - 7r_1^p\Big)r_{i,x}^p
  -(7y^2 +  6u)r_{i,y}^p
  \\
  &    +(i+8)r_{i+2}^p + 2y(p+2i+13)r_i^p + x(3p+5i+28)r_{i-1}^p
    \\
  &
  +\frac{p+1}{p+3}\,\left(2P_{0,i-2}^{(4)} +
    \sum_{j=1}^\infty\left(-\frac{1}{p+3}\right)^j 
    \cdot \prod_{l=0}^j(l(p+3)+2)\cdot P_{j,i-2j+2}^{(2)}\right).\end{aligned}$$ Explicit formulas for other nonlocal symmetries can be obtained using commutator relations from the proof of Theorem \[thm:lie-algebra-nonlocal\]. For example, the symmetry $\Phi_1$ that generates the positive part of $\operatorname{sym}_{\tau^p}\mathcal{E}_1$ is $$\Phi_1=(\phi_1^p,\phi_1^{1,p},\dots,\phi_1^{i,p},\dots)$$ with $$\phi_1^p=-\frac{6}{p+3}\,r_1^p -\big((4y^2+5u)u_x + 4xu_y - 2(3p+16)xy\big)$$ and $$\begin{aligned}
  \phi_1^{i,p}&=
  -\big((4y^2+5u)r_{i,x}^p + 4xr_{i,y}^p - (i+6)r_{i+1}^p -
  y(2p+3i+16)r_{i-1}^p\big)
  \\
  &
  +\frac{p+2}{p+3}\,\left(P_{0,i-1}^{(3)} + \sum_{j=1}^\infty
    \left(-\frac{1}{p+3}\right)^j\cdot\prod_{l=0}^j(l(p+3)+1) \cdot
    P_{j,i-2j+1}^{(2)}\right).\end{aligned}$$

Equation  {#sec:equation-eq:3}
---------

The results on this equation and their proofs are almost identical to those on Equation . So, we confine ourselves with the final description of $\operatorname{sym}_{\sigma^p}\mathcal{E}_2$. Introduce the weights $${\left|x\right|} = 2,\qquad {\left|y\right|} = 1,\qquad {\left|u\right|} = 3.$$ Consequently, the nonlocal variables in the covering $\sigma^p$ acquire the weights ${\left|r_i^p\right|} = i+3$.

Then we have

\[thm:equation-eqrefeq:3\] The symmetries $\gamma_{-2}$ and $\gamma_{-1}$ can be lifted to symmetries $\Gamma_{-2}$ and $\Gamma_{-1}$ in any covering $\sigma^p$ over Equation . In addition there exists a nonlocal symmetry $\Gamma_2$ of weight $2$. These three symmetries generate the entire Lie algebra $\operatorname{sym}_{\sigma^p}\mathcal{E}_2$ which is isomorphic to the Witt algebra $\mathfrak{W}$.

The symmetry $\Gamma_{-2} =
  (\gamma_{-2},\gamma_{-2}^{1,p},\dots,\gamma_{-2}^{i,p},\dots)$ is given by $\gamma_{-2} = -y-u_x/2$ and $$\gamma_{-2}^{i,p} = -\frac{1}{2}\left(r_{i,x}^p + (p+5)\left(r_{i-2}^p +
        \sum_{j=1}^\infty \left(-\frac{1}{p+3}\right)^j \cdot \prod_{l=0}^{j-1}
          (l(p+3) - 2)\cdot P_{j-1,i-j}^{(1)}\right)\right).$$ For $\Gamma_{-1} =
  (\gamma_{-1},\gamma_{-1}^{1,p},\dots,\gamma_{-1}^{i,p},\dots)$ we have $\gamma_{-1}= y^2 - 2 x + 2 yu_x - 2 u_y$ and $$\gamma_{-1}^{i,p} = 2\left(yr_{i,x}^p - r_{i,y}^p - (p+4)\left(r_{i-1}^p +
        \sum_{j=1}^\infty \left(-\frac{1}{p+3}\right)^j \cdot
        \prod_{l=0}^{j-1} (l(p+3)-1)\cdot P_{j-1,i-j+1}^{(1)}\right)\right).$$ Finally, we have $$\begin{aligned}
    \gamma_2^p &= 7r_2^p + \big((7p + 37)y-6u_x\big)r_1^p\\
    &+(p + 3) \left( (6p + 31)yu + (3p + 19)x^2 - (3p^2 + 18p + 13)xy^2 +
      \frac{1}{12}(3p^3+ 27p^2 + 81p + 89)y^4\right.\\
    &\left.  + \left(5u + 14xy + \frac{19}{3}y^3\right)u_y - \frac{1}{2}(7p^2+
      74p + 197)y^2 u - (7p + 37)xu - 4(2p +
      7)x^2y\right. \\
    &\left.+ \frac{1}{3}(7p^3+ 87p^2 + 333p + 397)xy^3 - \frac{1}{30}(7p^4+
      104p^3 + 558p^2 + 1296p + 1115)y^5\right)
  \end{aligned}$$ and $$\gamma_2^{i,p}=-(p+3)((i + 7)r_{i+2}^p + (p + 3i + 19)yr_{i+1}^p) +
    Ar_{i,x}^p + Br_{i,y}^p + Cr_i^p +D$$ for $\Gamma_2 = (\gamma_2^p,\gamma_2^{1,p},\dots,\gamma_2^{i,p},\dots)$, where $$\begin{aligned}
    A&=(p + 3)\left(
      (6p + 31)uy + (3p + 19)x^2 - (3p^2 + 18p + 13)xy^2 +
      \frac{1}{12}(3p^3+ 27p^2 + 81p + 89)y^4\right)\\
    &- 6r_1^p,\\
    B&= (p + 3)\left(5u + 14xy + \frac{19}{3}y^3\right),\\
    C&=- (p + 3)\Big( 2(p + 2i + 11)x + (p^2+ 9p + 5i + 33)y^2 \Big),\\
    \intertext{and}
    D&=-(p + 1)\left(2P_{0,i-1}^{(3)} + \sum_{j=1}^\infty
      \left(-\frac{1}{p+3}\right)^j\cdot
      \prod_{l=0}^j(l(p + 3) + 2)\cdot P_{j,i-j+3}^{(1)}
    \right)
  \end{aligned}$$ for $i\geq 1$.

Discussion {#sec:discussion}
==========

We conclude with two remarks.

\[sec:discussion-1\] Consider the covering  with $p=2$ and the generating series $$R=\sum_{i=1}^\infty\lambda^ir_i$$ of the corresponding nonlocal variables. The function $R$ satisfies the system $$\begin{aligned}
    R_x&=\lambda(R_y + u_xR_x),\\
    R_y&=\lambda(u_y+y)R_x - (\lambda^3R)_\lambda,
  \end{aligned}$$ which is equivalent to $$\label{eq:14}
    \begin{array}{lcr}
    \tilde{R}_x&=\dfrac{\lambda^4\tilde{R}_\lambda}{\lambda^2(u_y+y) +
      \lambda u_x - 1},\\[10pt]
    \tilde{R}_y&=\dfrac{\lambda^3(1-\lambda
      u_x)\tilde{R}_\lambda}{\lambda^2(u_y+y) + \lambda u_x - 1},
  \end{array}$$ where $\tilde{R} = \lambda^3R$. Let $\tilde{R}=\tilde{R}(x,y,\lambda)$ be a solution to such that $\tilde{R}_\lambda \not \equiv 0$. Then equation $\tilde{R}(x,y,\lambda) =\mathrm{const}$ defines $\lambda$ as a fuction of $x$ and $y$, that is, $\lambda = \psi(x,y)$, cf. [@Pavlov-2009]. Then $$\tilde{R}_\lambda\cdot\psi_x + \tilde{R}_x = 0,\quad
    \tilde{R}_\lambda\cdot\psi_y + \tilde{R}_y = 0$$ and  transforms to $$\begin{array}{lcr}
      \psi_x&=&\dfrac{\psi^4}{1-u_x\psi-(u_y+y)\psi^2},\\[10pt]
      \psi_y&=&\dfrac{\psi^3(1-u_x\psi)}{1-u_x\psi-(u_y+y)\psi^2},
    \end{array}$$ which, by the gauge transformation $\psi\mapsto\psi^{-1}$, is equivalent to $$\begin{array}{lcr}
      \psi_x&=&-\dfrac{1}{\psi^2-u_x\psi-(u_y+y)},\\[10pt]
      \psi_y&=&\dfrac{u_x-\psi}{\psi^2-u_x\psi-(u_y+y)}.
    \end{array}$$ This is the covering obtained in [@B-K-M-V-2015] by the direct reduction and coincides with the known covering over the Gibbons-Tsarev equation, see [@Gibbons-Tsarev-1996].

\[sec:discussion-2\] In [@B-K-M-V-2014], two other equations, $$u_y u_{xy} - u_x u_{yy} = e^y u_{xx}$$ and $$u_{yy} = (u_x + x)u_{xy} - u_y (u_{xx} + 2),$$ were obtained as symmetry reductions of the universal hierarchy and the 3D rdDym equations, respectively, were obtained. We plan study these equations by the methods similar to the used above in the forthcoming publications.

Acknowledgments {#acknowledgments .unnumbered}
===============

Computations were supported by the <span style="font-variant:small-caps;">Jets</span> software, [@Jets].

The first author (PH) was supported by the Specific Research grant SGS/6/2017 of the Silesian University in Opava. The third author (OIM) is grateful to the Polish Ministry of Science and Higher Education for financial support.

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[^1]: We do not distinguish between scalar and multi-component systems.

[^2]: The case $p=-3$ is special, but, as it was indicated above, is reduced to the case $p=-2$.
