Consider the rYME (1), where x, y, z, t are the independent variables and u stands for the dependent variable. The manifold \(}\) corresponding to this equation lies in the jet space \(J^(\pi )\), where \(\pi :}\times }^4\rightarrow }^4\). Then the internal coordinates on \(}\) are \(u_,u_,u_\) where \(i,j\ge 0,\ k,l>0\), and the Cartan distribution on \(}\) is spanned by the total derivative operators \(D_x,D_y,D_z,D_t\) on \(J^(\pi )\) restricted to \(}\), that is

As it has already been mentioned in the introductory part of this paper, the rYME (1) was introduced for the first time in [12], including its one-parameter isospectral Lax representation

$$\begin s_t=u_zs_x+\varkappa s_z, \quad s_y=(u_x+\varkappa )s_x, \end$$

(12)

cf. also [9]. However, it turns out that the Lax pair (12) can be easily modified by the substitution \(\varkappa \rightarrow -\frac\) into the Lax pair which contains two parameters and explicitly depends on the independent variables t, z. Indeed, it is straightforward to verify that the equations

$$\begin w_t=u_zw_x-\frac\,w_z, \quad w_y=u_xw_x-\frac\,w_x, \end$$

(13)

where \(\lambda ,\mu \in }\) are spectral parameters and w is a nonlocal variable, also define the Lax representation of the rYME (1).

Remark 1

In fact, the “nonlocal variables” produced by Lax pairs for equations in more than two independent variables are not nonlocal variables in the sense of the definition stated in Sect. 1. However, as we will discuss below, these “nonlocal variables” provide us with an infinite number of nonlocal variables for the given equation. Thus, until we analyze this problem for the case of the rYME (1) in more details, we will call w and other variables given by the Lax pairs for \(}\) to be just nonlocal variables.

To obtain coverings of (1), we consider the following scheme applied before, for example, in [3, 16, 32], cf. also [23, 30]:

(1)

The nonlocal variable w given by the Lax pair (13) is expanded in a formal series in the spectral parameter \(\lambda \), resp. \(\mu \):

$$\begin w=\sum \limits _^\infty \lambda ^i w_i, \; \mathrm \quad w=\sum \limits _^\infty \mu ^i w_i. \end$$

(2)

By substituting this expansion into the Lax pair (13) and eliminating the parameter \(\lambda \), resp. \(\mu \), we get an infinite series of nonlocal quantities \(w_i\), \(i \in }\).

(3)

To achieve the proper definition of nonlocal variables \(w_i\), we consider two reductions of the series \(w_i\): (a) the negative reduction \(w_i=0\) for \(i>0\) and (b) the positive reduction \(w_i=0\) for \(i<0\).

(4)

In this way, we arrive at the following three inequivalent infinite-dimensional coverings \(\tau ^q\), \(\tau ^m\) and \(\tau ^r\) (given in the form of Lax pairs) with the nonlocal variables \(q_\alpha ,m_\alpha ,r_\beta \), respectively, where

$$\begin \tau ^q:\tilde}}^q\rightarrow }&\quad \left| \begin q_ = 0,\\ q_ = u_zq_ + q_,\\[2pt] q_ = u_xq_ + q_, \quad \alpha \ge 0; \end\right. \end$$

(14)

$$\begin \tau ^m:\tilde}}^m\rightarrow }&\quad \left| \begin m_ = 0,\\ m_ = u_z m_-\displaystyle \fracm_-\fracm_,\\[3mm] m_ = u_x m_-\displaystyle \fracm_-\fracm_, \quad \alpha \ge 0; \end\right. \end$$

(15)

and

$$\begin \tau ^r:\tilde}}^r\rightarrow }&\quad \left| \begin r_ = x,\quad r_0=-u\\ r_ = u_xr_ - r_,\\[2pt] r_ = u_zr_ - r_, \quad \beta \ge 1. \end\right. \end$$

(16)

Remark 2

Let us note that the coverings \(\tau ^q\) and \(\tau ^m\) are non-Abelian, whereas \(\tau ^r\) is Abelian. Thus, the nonlocal variables \(r_\beta \), \(\beta \ge 1\), are interconnected with infinite series of two-component nonlocal conservation laws of the rYME (1).

Remark 3

Whereas the coverings \(\tau ^q\) and \(\tau ^r\) can also be derived from the one-parameter Lax pair (12), the covering \(\tau ^m\) can be obtained only from the two-parameter Lax representation (13).

As noted in Remark 1, Eqs. (14), (15) and (16) are not in the form of the covering Eq. (8). In order to identify the nonlocal coordinates provided by these formulas, we have to introduce the infinite number of formal variables \(q_^\), \(m_^\) and \(r_^,\) \(a,b,=0,1,\dots \), where \(q_^=q_\), \(m_^=m_\), \(r_^=r_\) and

$$\begin q_^= & q_^, \quad q_^=q_^,\quad q_^=(u_xq_^ + q_^)_z^b},\\ q_^= & (u_zq_^ + q_^)_z^b},\\ m_^= & m_^,\quad m_^=m_^,\quad m_^=\left( u_x m_^-\displaystyle \fracm_^-\fracm_^\right) _z^b},\\ m_^= & \left( u_z m_^-\displaystyle \fracm_^-\fracm_^\right) _z^b},\\ r_^= & r_^,\quad r_^=r_^,\\ r_^= & (u_xr_ - r_^)_,\quad r_^=( u_zr_ - r_^)_. \end$$

Note that all the right-hand sides of the equalities above are (even though sometimes inductively) expressible in terms of the variables \(q_^\), \(m_^\) and \(r_^,\) thus, they provide us with the covering equations \(}_\), \(}_\) and \(}_\) in the form (8), \(q_^\), \(m_^\) and \(r_^,\) \(\alpha =0,1,\dots ,\beta =1,2,\dots \), being the true nonlocal variables.

The total derivative operators in the covering \(\tau ^q\) now take the form (cf. (6))

$$\begin }_x^=}}_x\!+\!X^,\!\quad }_y^=}}_y\!+\!Y^,\!\quad }_z^=}}_z\!+\!Z^,\!\quad }_t^=}}_t+T^, \end$$

where

$$\begin X^&=\sum _^\sum _^q_^\frac^},\qquad&Y^=\sum _^\sum _^(u_xq_ + q_)_z^b}\frac^},\\ Z^&=\sum _^\sum _^q_^\frac^},&T^=\sum _^\sum _^(u_zq_ + q_)_z^b}\frac^}. \end$$

The total derivative operators \(D_x^\), \(D_x^\), \(D_y^\), \(D_y^\), etc. in the coverings \(\tau ^m\) and \(\tau ^r\) are given in a similarly way.

In what follows, we will consider the Whitney product \(\tau ^W=\tau ^q \oplus \tau ^m \oplus \tau ^r\) of all three coverings and carry out all calculations in \(\tau ^W\). The covering space \(}}^W=}_\cap }\) can be considered to be an equation in the jet space \(J^(}})\) given by the rYME (1) and Eqs. (14)–(16) for \(\alpha \ge 0\), \(\beta \ge 1\), the coordinates in \(J^(}})\) being \(x,y,z,t,u_,q_^,m_^,r_^,\ a,b\ge 0\). The Cartan distribution on \(\tilde}}^W\) is spanned by the total derivative operators

$$\begin }}_x&=}}_x+X^+X^+X^,&}}_y&=\bar_y+Y^+Y^+Y^,\\ }}_z&=}}_z+Z^+Z^+Z^,&}}_t&=\bar_t+T^+T^+T^. \end$$

As follows directly from (9)–(11), every full-fledged nonlocal symmetry of \(}\) in the Whitney product \(\tau ^W\) (for shortness we will write only \(\tau ^W\)-symmetry below) is of the form

$$\begin & }_P =}^W_+\sum _^\sum _^ \\ & \quad \left( }}_x^a}}_z^b(p_^)\frac^} +}}_x^a}}_z^b(p_^)\frac^}+}}_y^a}}_t^b(p_^)\frac^}\right) , \end$$

where

and \(p_0, p_^,p_^, p_1^, p_^\), \(\alpha \ge 0,\beta \ge 2\), are smooth functions on \(\tilde}}^W\) satisfying the conditions:

$$\begin }}_(p_0) - }}_(p_0) + u_z }}_(p_0)-u_x}}_(p_0)+u_}}_z(p_0)-u_}}_x(p_0)&=0, \end$$

(NS1)

$$\begin }}_t(p_\alpha ^q) - u_z}}_x(p_\alpha ^q) - q_}}_z(p_0)-}}_z(p_^q)&=0, \end$$

(NS2)

$$\begin }}_y(p_\alpha ^q) -u_x}}_x(p_\alpha ^q) - q_}}_x(p_0)-}}_x(p_^q)&=0, \end$$

(NS3)

$$\begin }}_t(p_\alpha ^m) -u_z}}_x(p_\alpha ^m) - m_}}_z(p_0)+\frac}}_z(p_\alpha ^m)+\frac}}_z(p_^m)&=0, \end$$

(NS4)

$$\begin }}_y(p_\alpha ^m) -u_x}}_x(p_\alpha ^m) - m_}}_x(p_0)+\frac}}_x(p_\alpha ^m)+\frac}}_x(p_^m)&=0, \end$$

(NS5)

$$\begin }}_x(p_1^r) - }}_y(p_0)+2u_x}}_x(p_0)&=0, \end$$

(NS6)

$$\begin }}_z(p_1^r) -}}_t(p_0)+u_x}}_z(p_0)+u_z}}_x(p_0)&=0, \end$$

(NS7)

$$\begin }}_x(p_\beta ^r) - u_x}}_x(p_^r)-r_}}_x(p_0)+D_y(p_^r)&=0, \end$$

(NS8)

$$\begin }}_z(p_\beta ^r) - u_z}}_x(p_^r)-r_}}_z(p_0)+D_t(p_^r)&=0, \end$$

(NS9)

where \(p_^q=p_^m=0\).

Thus, any \(\tau ^W\)-symmetry can be (uniquely) represented by the vector-valued generating function \(P=\left[ p_0, p_0^q, p_0^m, p_1^r,p_1^q, p_1^m, p_2^r,p_2^q, p_2^m\ldots \right] \). From now on, unless otherwise stated, by a nonlocal symmetry of \(}\) we mean a generating function of a \(\tau ^W\)-symmetry of \(}\), and \(\textrm^(})\) will denote the set of all generating functions of \(\tau ^W\)-symmetries of \(}\) (equipped with the Lie algebra structure given by the Jacobi bracket).

Remark 4

Let us note that we have used the formal nonlocal variables \(q_^,m_^,r_^\) in order to justify the general form of the (generating function of) \(\tau ^W\)-symmetry of \(}\). However, since the equalities \(q_\equiv q_^=q_^,m_\equiv m_^=m_^,r_\equiv r_^=r_^ \) hold on \(\tilde}}^\), we will use the more common notation \(q_,m_,r_\) for them. Let us also emphasize that we sometimes use terms like \(m_, r_\), etc., in the explicit formulas presented below, even though \(m_, r_\), etc., are not internal coordinate on \(\tilde}}^W.\) In fact, such terms are used to rewrite the huge resulting formulas into a more concise form.

The task to find a recursion operator for \(\tau ^W\)-symmetries for \(}\) now coincides with the problem to find a Bäcklund auto-transformation of the tangent equation \(} \tilde}}^W\) given by Eqs. (1) and (NS1)–(NS9) with the new dependent variables \(p_,p_^q, p_^m, p_^r\), the tangent covering being

$$\begin&t^W: } }}}^W \rightarrow }}^W,\\&\quad (x,y,z,t,u_\sigma ^j,q_,m_, r_, p_,p_^q, p_^m, p_^r)\\&\qquad \mapsto (x,y,z,t,u_\sigma ^j,q_,m_,r_). \end$$

The most straightforward way is to look for a Bäcklund auto-transformation expressible in the terms of the existing variables, i.e., to consider the covering equation common for both the copies of \(}}}}^W\) to be just \(}}}}^W\) itself. Due to the \(}\)-linearity requirement imposed on the sought operator, we look for the desired relationship between the solutions to \(}}}}^W\) in the following form: the components of the ‘new’ solution to \(}}}}^W\) are to be (finite) linear combinations of the components \(p_,p_^q, p_^m, p_^r\) of the ‘old’ solution to \(}}}}^W\), the coefficients being arbitrary functions of the internal coordinates on \(}}^W\). This ‘symmetry’, if it exists and we are able to describe it in an usable form, provides us with a recursion operator for \(\tau ^W\)-symmetries of \(}\).

The issue described above can be executed in the following way: we suppose that the quantity \(P=\left[ p_0, p_0^q, p_0^m, p_1^r,p_1^q, p_1^m, p_2^r,p_2^q, p_2^m,\ldots \right] \) is a \(\tau ^W\)-symmetry of the rYME (1), i.e., the components of P satisfy the conditions (NS1)–(NS9) on \(}}}^W\), and subsequently try to solve Eqs. (NS1)–(NS9) with respect to the unknown quantity \(}}=\left[ }}_0, }}_0^q, }}_0^m, }}_1^r,}}_1^q, }}_1^m, }}_2^r, }}_2^q, }}_2^m,\ldots \right] \) under the assumption that the components of \(}}\) are linear combinations of the components of P and their derivatives up to some finite order, the coefficients being (possibly) arbitrary functions of the internal coordinates on \(\tilde}}^W\).

Note that even though the idea of how to find a full-fledged recursion operator is quite straightforward, the implementation of the latter is not simple at all. The main difficulty of this construction consists in the fact that the recursion operator we are looking for is to act on an infinite sequence resulting in another infinite sequence, so that our success depends on whether we are able to find general formulas describing the relationships between all the components of the pre-image and its image.

One but not the only one result (see Sect. 3 below) we obtained within this approach is the following:

Proposition 1

Let \(P=\left[ p_0, p_0^q, p_0^m, p_1^r,p_1^q, p_1^m, p_2^r,p_2^q, p_2^m,\ldots \right] \) be a \(\tau ^W\)-symmetry of the rYME (1), and let \(}}=\left[ }}_0, }}_0^q, }}_0^m, }}_1^r,}}_1^q, }}_1^m, }}_2^r, }}_2^q, }}_2^m,\ldots \right] \) be a vector-valued function such that for each \(\alpha \ge 0, \beta \ge 1\) it holds

$$\begin }}_0&= u_x p_0+p_1^r, \end$$

(17)

$$\begin }}_\alpha ^q&= q_ p_0 + p_^q, \end$$

(18)

$$\begin }}_\alpha ^m&= m_ p_0 -\frac p_^m - \frac p_\alpha ^m , \end$$

(19)

$$\begin }}_\beta ^r&= r_ p_0 - p_^r. \end$$

(20)

Then \(}}\) is also a \(\tau ^W\)-symmetry of the rYME (1).

Proof

The proof is done by straightforward computations and is presented in Appendix A. \(\square \)

Remark 5

According to Proposition 1, Eqs. (17)–(20) provide us with a relation which maps any known \(\tau ^W\)-symmetry P to a new \(\tau ^W\)-symmetry \(}}\). Hence, they can be viewed as the determining equations of a recursion operator for \(\tau ^W\)-symmetries of the rYME (1). In what follows, we will denote this recursion operator as \(}^q\). The notation will be clarified a bit later in Remark 10, see Sect. 3 below.

Remark 6

One can also easily observe that Eqs. (17)–(20) can be rewritten in the form

$$\begin }}}^T=}^q\cdot P^, \end$$

where \(^T\) denotes the transpose of the row vectors P and \(}}\), respectively, \(\cdot \) is the usual matrix multiplication, and the recursion operator \(}^q\) is considered to be an infinite-dimensional matrix whose structure is illustrated by its \(13 \times 16\) left-upper corner as follows:

$$\begin }^q}= \left( \begin u_x & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\ldots \\ q_ & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\ldots \\ m_ & 0 & -z/t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\ldots \\ r_ & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\ldots \\ q_ & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &0 & 0 &\ldots \\ m_ & 0 & -1/t & 0 & 0 & -z/t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\ldots \\ r_ & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 &\ldots \\ q_ & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\ldots \\ m_ & 0 & 0 & 0 & 0 & -1/t & 0 & 0 & -z/t & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\ldots \\ r_ & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 &\ldots \\ q_ & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\ldots \\ m_ & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/t & 0 & 0 & -z/t & 0 & 0 & 0 & 0 &\ldots \\ r_ & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 &\ldots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end\right) \end$$

As it can be easily verified, each column of the matrix \(}^q\) is itself a \(\tau ^W\)-symmetry of the rYME (1). However, this is perfectly natural, because if the matrix \(}^q\) is supposed to represent a recursion operator, then the images of the simplest nontrivial \(\tau ^W\)-symmetries \((1,0,0,\ldots ), (0,1,0,\ldots ),\) etc. must again be \(\tau ^W\)-symmetries.

Remark 7

Let us also note that using Eqs. (NS6)–(NS7) we can readily eliminate the quantity \(p_1^r\) from Eq. (17). In this way, we obtain the conventional form (i.e., the form most frequently used in the current literature, cf., e.g., [2, 3, 16, 31, 32]) of the recursion operator \(}^q_\) for shadows of \(\tau ^W\)-symmetries of the rYME (1), which reads

$$\begin \begin }^q_: }}_&= p_-u_x p_ + u_ p_0,\\ }}_&= p_-u_z p_ + u_ p_0. \end \end$$

(21)

As far as we know, even this recursion operator for shadows has not been, quite surprisingly, presented in literature yet.