## Abstract

We present a deautonomization procedure for partial difference and differential-difference equations (with the latter defining symmetries of the former) which uses the integrability conditions as integrability detector. This procedure is applied to Hirota’s Korteweg–de Vries and all the ABS equations and leads to non-autonomous equations and their non-autonomous generalized symmetries of order two, all of which depend on arbitrary periodic functions and are related to the same two-quad equation and its symmetries. We show how reductions of the derived differential-difference equations lead to alternating QRT maps, and periodic reductions of the difference equations result to non-autonomous maps and discrete Painlevé type equations.

Communicated by: Prof. Nalini Joshi

## 1. Introduction

There exist several criteria and methods to test the integrability of a given difference equation and to derive discrete integrable systems. Multidimensional consistency [1], algebraic entropy [2], singularity confinement [3] and integrability conditions (existence of infinite hierarchies of generalized symmetries and canonical conservation laws) [4, 5] are some of these criteria and, for instance, multidimensional consistency has been used to classify integrable equations in [6, 7], and singularity confinement has been employed for the deautonomization of known autonomous equations [3, 810].

These criteria apply to any difference equation and there is a plethora of autonomous equations which satisfy them and can be characterized as integrable. For instance, the discrete potential Korteweg–de Vries (KdV) equation, aka H1, $$(u_{n,m}-u_{n+1,m+1}) (u_{n+1,m}-u_{n,m+1}) = \alpha - \beta$$ is multidimensionally consistent [6], it admits infinite hierarchies of symmetries in both directions [11], its movable singularities are confined [3] and its algebraic entropy vanishes [12]. Moreover, its deautonomized counterpart
$$(u_{n,m}-u_{n+1,m+1}) (u_{n+1,m}-u_{n,m+1}) = \alpha_{n} - \beta_m$$
(1)
is also multidimensionally consistent, with confined singularities [13] and vanishing algebraic entropy [12] but it does not admit any generalized symmetries for generic functions $$\alpha_{n}$$ and $$\beta_{m}$$ but only for specific choices of these two functions [14]. So the question we try to answer here is if we can systematically specify any arbitrary functions of the independent variables in a given deautonomized equation so that the resulting equation to admit hierarchies of symmetries and canonical conservation laws in both lattice directions as its autonomous counterpart.

To answer this question, we are going to employ the integrability conditions [4, 5] which not only test the integrability of the equation under consideration but also serve as determining equations for its generalized symmetries. So this approach allows us not only to derive integrable non-autonomous partial difference equations but also to construct their symmetries (differential-difference equations) and conservation laws which will also depend explicitly on $$n$$ and $$m$$.

Specifically we start with known integrable partial difference equations in a generic deautonimized form and employ the integrability conditions to separate the integrable cases, i.e. the ones admitting generalized symmetries and conservation laws in both lattice directions. For our purposes, it is sufficient to consider only the first of these conditions which for the symmetries in the $$n$$ direction can be written as
$$\left( {\cal{T}}-1 \right)(\ln R) = ({\cal{S}}^N-1) {\cal{S}}^{-d}\left(\ln \frac{\partial_{u_{n+d,m+1}} Q_{n,m}}{\partial_{u_{n+d,m}} Q_{n,m}}\right)\!.$$
(2)

Here, $${\cal{S}}$$ and $${\cal{T}}$$ are the shift operators in the first and second direction, respectively, $$Q_{n,m}$$ stands for the defining function of the $$d$$-quad equation under consideration, and $$R$$ is a function of $$n$$, $$m$$ and variables $$u_{n-N,m}$$, $$\ldots$$, $$u_{n+N,m}$$.

As $$N$$ depends on the order of the symmetry (i.e. the number of forward and backwards shifted values of $$u$$ involved in the symmetry generator) and there is no way to choose $$N$$ a priori, we start our investigations with $$N=1$$ (first order symmetries) and then proceed to second order ones ($$N=2$$). In every step $$N$$, we separate the integrable equations as the ones admitting symmetries of order $$N$$ and none of lower order (when $$N>1$$). This procedure determines completely the introduced functions of $$n$$ and $$m$$. Finally, employing more of the integrability conditions, we construct the lowest order symmetries which are non-autonomous differential-difference equations of order two.

We applied this procedure to equations (1), and the deautonomized Hirota’s KdV [15]
$$v_{n,m} + \frac{a_{n,m+1}}{v_{n,m+1}} - \frac{a_{n+1,m}}{v_{n+1,m}} - v_{n+1,m+1} = 0,$$
(3)
as well as to the ABS list, the sine-Gordon and the Hietarinta–Viallet equations (see (31) and (33) below). The only integrable cases for $$N=1$$ are the original autonomous equations. For symmetries of order two, it turns out that the arbitrary functions must be periodic with period two.
The most interesting phenomenon is that most of these deautonomized equations, we analysed are related via Miura transformations to the same non-autonomous quadratic one-quad equation (see equation (6) below) which can be autonomized leading to the two-quad equation
\begin{eqnarray} && z_{n+1,m} z_{n+1,m+1} \left(\phi_{n,m+1} -\phi_{n,m} \right) - z_{n+1,m+1} z_{n+2,m} (z_{n+1,m}+1) \left(\phi_{n,m} + z_{n,m} z_{n+1,m+1}\right) \nonumber\\ &&\quad +\,z_{n+1,m} z_{n+2,m+1} (z_{n+1,m+1}+1) \left(\phi_{n,m+1} + z_{n+1,m} z_{n,m+1}\right) = 0, \end{eqnarray}
(4)
where $$\phi_{n,m} = z_{n,m}+z_{n+1,m} + z_{n,m} z_{n+1,m}$$. Moreover, under the same Miura transformation the symmetries of the aforementioned quad equations are transformed to symmetries of the two-quad equation (4). In particular, the first symmetry is given by the differential-difference equation
\begin{align} \partial_{t} z_{n,m} &= c_n (z_{n,m}+1) \left(\frac{z_{n,m} (z_{n+1,m}+1) z_{n+2,m}}{z_{n+1,m}} - \frac{z_{n,m} (z_{n-1,m}+1) z_{n-2,m}}{z_{n-1,m}}\right)\notag\\ &\quad +\, c_{n+1} (z_{n,m}+1) (z_{n+1,m}-z_{n-1,m}), \end{align}
(5)
where $$c_n$$ is a periodic function with period 2, i.e. $$c_{n+2} = c_n$$. This is an integrable equation not only because it satisfies the corresponding integrability conditions but also because it admits a local master symmetry (see (10) below). Moreover (5) may be viewed as a generalization of the differential-difference equation presented recently by Adler [16] and corresponds to the choice $$c_n=1$$. The fact that all these second order differential-difference equations we found are mapped to (5) suggests that the latter plays a role of universal object in the theory as it was pointed in [17].
Since all the second order differential-difference equations, we derive depend on periodic functions of $$n$$, certain reductions of them lead to alternating QRT maps [18, 19] with first integrals following from the conserved forms of the equations involved. On the other hand, $$(k,-1)$$ periodic reductions of the non-autonomous partial difference equations lead to families of ordinary difference equations the form of which depends on the parity of $$k$$. In particular, the even order reductions of (3) with $$a_{n,m} = \alpha_{n}$$ are non-autonomous extensions of some of the maps discussed recently in [20], and are related via a Bäcklund transformation to the corresponding reductions of (1) with $$\beta_{m}=0$$. But the odd order reductions of equation (3) lead to the family of equations
$$\prod_{i=0}^{p} x_{n+2 i} - \prod_{i=0}^{p-1} x_{n+2 i + 1} = b_n,\,\, {\mbox{ where }}\,\, b_{n+2} = b_n,$$
the first member ($$p=1$$) of which coincides with the first two-periodic Lyness recurrence [21], and the corresponding reductions of (1) result to equations
$$x_{n} + \cdots + x_{n+2 p} - (-1)^p x_{n+2 p} \prod_{i=0}^{p-1} \frac{x_{n+2 i}}{x_{n+2 i+1}}\, = \alpha \,n + \beta (-1)^n + \gamma,$$
where $$\alpha$$, $$\beta$$ and $$\gamma$$ are real constants.

The article is organized as follows. Section 2 presents the non-autonomous quadratic quad equation and its autonomous two-quad counterpart and some of their integrability aspects. Section 3 deals with the procedure of deautonomization using Hirota’s and the discrete potential KdV equations as illustrative examples and presents the relative results for all the ABS equations, the deformed H list, as well as for the discrete sine-Gordon and the Hietarinta-Viallet equations. Section 4 presents reductions for some of the derived non-autonomous differential-difference equations and their relations to alternating QRT maps. Section 5 deals with the $$(k,-1)$$ periodic reductions of the deautonomized KdV equations. The final section gives an overall evaluation of our results and the appendix contains outcomes from our analysis in Section 3 and some canonical conservation laws we employ in section 5.

## 2. Two difference equations and their symmetries

In this section, we present two difference equations, derive their relation and discuss their integrability properties. In particular, we present their lowest order generalized symmetries in the first lattice direction (of order two) and compute the first two canonical conserved densities. The importance of these two equations stems from the fact that both of them are related to the deautonomized equations we are discussing in the next section.

The first equation is non-autonomous, quadratic and defined on an elementary quadrilateral of the lattice and has the following form.

\begin{align} W_{n,m}(A_n,A_{n+1},B) &:= A_n (\phi_{n,m}+z_{n,m} z_{n+1,m+1}) (\phi_{n,m+1}+ z_{n+1,m} z_{n,m+1}) \, \nonumber \\ &\quad - A_{n+1} (\phi_{n,m}+z_{n+1,m} z_{n,m+1}) (\phi_{n,m+1}+ z_{n,m} z_{n+1,m+1}) \nonumber\\ &\quad -B (\phi_{n,m+1} - \phi_{n,m})(z_{n,m} z_{n+1,m+1} - z_{n+1,m} z_{n,m+1}) =0, \end{align}
(6a)
where
$$\phi_{n,m} = z_{n,m}+z_{n+1,m} + z_{n,m} z_{n+1,m},$$
(6b)
and $$A_{n+2}= A_n$$, $$A_{n+1} \ne A_n$$ and $$B$$ is a constant. The second condition for $$A_n$$ is essential because if $$A_n= A \in {\mathbb{R}}$$, then equation is linearizable since $$W_{n,m}(A,A,B) = (A-B) (\phi_{n,m+1} - \phi_{n,m})(z_{n,m} z_{n+1,m+1} - z_{n+1,m} z_{n,m+1})$$.
An interesting property of this equation is that we can remove function $$A_n$$ from our considerations using its periodicity and the pair of equations $$W_{n,m}(A_n,A_{n+1},B) = 0$$, $$W_{n+1,m}(A_{n+1},A_n,B) = 0$$. The elimination of $$A_n$$ and $$A_{n+1}$$ from this pair leads to a polynomial equation that can be factorized as a product of two polynomials, each of which is linear in $$z_{n,m}$$, $$z_{n,m+1}$$, $$z_{n+2,m}$$ and $$z_{n+2,m+1}$$. The first factor leads to
$$(\phi_{n,m}+z_{n,m} z_{n+1,m+1}) z_{n+2,m+1} - (\phi_{n,m+1}+ z_{n+1,m} z_{n,m+1}) z_{n+2,m} + z_{n,m} z_{n+1,m+1} - z_{n+1,m} z_{n,m+1} = 0,$$
(7)
which is a trivial equation since it is equivalent to $$({\cal{T}}-1) ({\cal{S}}^2-1) \log (z_{n,m}-z_{n-2,m}) = 0$$. The second factor however yields the two-quad equation
\begin{eqnarray} && z_{n+1,m} z_{n+1,m+1} \left(\phi_{n,m+1} -\phi_{n,m} \right) - z_{n+1,m+1} z_{n+2,m} (z_{n+1,m}+1) \left(\phi_{n,m} + z_{n,m} z_{n+1,m+1}\right) \nonumber\\ && \quad+\,z_{n+1,m} z_{n+2,m+1} (z_{n+1,m+1}+1) \left(\phi_{n,m+1} + z_{n+1,m} z_{n,m+1}\right) = 0. \end{eqnarray}
(8)

This is an integrable equation, as we explain below, and can be solved uniquely with respect to any corner value of $$z$$ (hence, we can solve uniquely the initial value problem if initial data are given for instance along one horizontal line and two consecutive vertical lines).

A manifestation of the integrability of equations (6) and (8) is that they admit the same hierarchy of generalized symmetries in the $$n$$ direction with the lowest symmetry being of order two. More precisely the first member of this hierarchy is given by
\begin{align} \partial_{t} z_{n,m} &= c_n (z_{n,m}+1) \left(\frac{z_{n,m} (z_{n+1,m}+1) z_{n+2,m}}{z_{n+1,m}} - \frac{z_{n,m} (z_{n-1,m}+1) z_{n-2,m}}{z_{n-1,m}}\right)\notag\\ &\quad +\, c_{n+1} (z_{n,m}+1) (z_{n+1,m}-z_{n-1,m}), \end{align}
(9)
where $$c_n$$ is a periodic function with period 2, i.e. $$c_{n+2} = c_n$$. We can construct the higher order members of this hierarchy by considering commutators with the local master symmetry
\begin{align} \partial_{\tau} z_{n,m} &= (z_{n,m}+1) \left(\frac{(n+2) z_{n,m} (z_{n+1,m}+1) z_{n+2,m}}{z_{n+1,m}} - \frac{(n-2) z_{n,m} (z_{n-1,m}+1) z_{n-2,m}}{z_{n-1,m}}\right. \notag \\ &\qquad\qquad\qquad\quad \left. + (n+1) z_{n+1,m}+ 2 z_{n,m} - (n-1) z_{n-1,m}\vphantom{\frac{(n+2)}{z_{n,m}}}\right)\!, \end{align}
(10)
which is another symmetry of equation (8) and was derived using the integrability conditions for equation (8). Moreover using the same conditions one may derive a hierarchy of canonical conservation laws and here we present (the equivalent to) the first two canonical conservation laws for (9).
First canonical conservation law
$$\rho^{(0)} = (-1)^{n} \log z_{n,m},\quad \sigma^{(0)} = (-1)^n \frac{z_{n-1,m}+1}{z_{n-1,m}} \frac{z_{n,m}+1}{z_{n,m}} (c_n z_{n-2,m} z_{n,m} - c_{n+1} z_{n-1,m} z_{n+1,m}),$$
(11)
where $$\partial_{t} \rho^{(0)} = \left({\cal{S}}-1\right) (\sigma^{(0)})$$.
Second canonical conservation law
\begin{eqnarray} \rho^{(1)} &=& z_{n,m} + z_{n+2,m} + 2 z_{n,m} z_{n+2,m} + \frac{2 z_{n,m} z_{n+2,m}}{z_{n+1,m}} \\ \end{eqnarray}
(12a)

\begin{eqnarray} \sigma^{(1)} &=& c_n z_{n,m} (z_{n,m}+1)\left(2 \frac{z_{n+1,m} +1}{z_{n+1,m}} \frac{z_{n-1,m}+1}{z_{n-1,m}} z_{n+2,m} z_{n-2,m} + \frac{z_{n+1,m}+1}{z_{n+1,m}} z_{n+2,m}+ \frac{z_{n-1,m}+1}{z_{n-1,m}} z_{n-2,m}\right) \nonumber \\ && +\, c_{n+1} (2 z_{n,m} + (z_{n,m}+1) (z_{n+1,m}+z_{n-1,m})), \end{eqnarray}
(12b)
where $$\partial_{t} \rho^{(1)} = \left({\cal{S}}^2-1\right) (\sigma^{(1)})$$.
For difference equation (8) the first canonical conservation law is equivalent to
$$\varrho^{(0)} = (z_{n+1,m}+1)^2,\quad \psi^{(0)} = \frac{(\phi_{n,m}+z_{n+1,m} z_{n,m+1}) (\phi_{n,m+1}+z_{n+1,m} z_{n,m+1})}{(\phi_{n,m}+z_{n,m} z_{n+1,m+1}) (\phi_{n,m+1}+z_{n,m} z_{n+1,m+1})},$$
with $$({\cal{T}}-1) \log \varrho^{(0)} = ({\cal{S}}-1) \log \psi^{(0)}$$, and for the second law the density $$\rho^{(1)}$$ is given in (12) but the flux is omitted here because of its length.

The symmetries in the vertical direction for the quadratic equation (6) can be derived by exploiting its relation to the quad equations presented in the next section. However, for equation (8) we do not know any symmetries in the $$m$$ direction.

## 3. Deautonomizations

Integrability conditions provide us the means not only to test if a given equation is integrable but also to find its generalized symmetries [5]. In this context, we employ equation (2) to find non-autonomous integrable generalizations of known partial difference equations defined on an elementary quadrilateral on the lattice. We describe our approach by using Hirota’s KdV and discrete potential KdV equations as illustrative examples and then present some results about the ABS equations [6] and their deformed counterparts [22].

### 3.1 Hirota’s KdV equation

$$v_{n,m} + \frac{a_{n,m+1}}{v_{n,m+1}} - \frac{a_{n+1,m}}{v_{n+1,m}} - v_{n+1,m+1}= 0$$
(13)
and try to find the functions $$a_{n,m}$$ for which it admits infinite hierarchies of symmetries in both lattice directions.
We start our analysis with the symmetries in the first direction in which case condition (2) with $$d=1$$ becomes
$${\cal{T}}(R) = \frac{a_{n,m} v_{n+N,m}^2}{a_{n+N,m} v_{n,m}^2} \,R,\quad R = R(n,m,v_{n-N,m},\ldots,v_{n+N,m}).$$
(14)
Assuming that equation (13) admits a first order formal recursion operator, the analysis of (14) with $$N=1$$ leads to a linear system of first order partial differential equations for $$R$$ which is consistent if and only if either $$R=0$$ or $$a_{n+1,m} = a_{n,m}$$. Hence, $$a_{n,m}$$ must be independent of $$n$$ for the equation to admit symmetries of order one in the $$n$$ direction. Working in the same way in the $$m$$ direction we find also $$a_{n,m+1}=a_{n,m}$$. Hence, the only member of the family (13) which satisfies the first order integrability conditions in both directions is Hirota’s KdV equation
$$v_{n,m} + \frac{\alpha}{v_{n,m+1}} - \frac{\alpha}{v_{n+1,m}} - v_{n+1,m+1}= 0,$$
(15)
which corresponds to $$a_{n,m}=\alpha$$.

To find the equations in the family (13) the lowest generalized symmetries of which are of order two (at least in the $$n$$ direction), we consider the determining equation (14) with $$N=2$$ along with the requirement $$a_{n+1,m} \ne a_{n,m}$$ to exclude any symmetries of order one. A linear system of differential equations for $$R$$ can be derived which is consistent provided that $$a_{n+2,m} =a_{n,m}$$. Moving to symmetries in the $$m$$ direction and assuming that they are of order one, then our previous analysis implies that $$a_{n,m+1}=a_{n,m}$$. However, assuming that the equation admits symmetries of order two in the $$m$$ direction, our analysis yields two constraints, namely $$a_{n,m+2}= a_{n,m}$$ and $$a_{n+1,m+1} + a_{n,m} = a_{n+1,m}+a_{n,m+1}$$, which imply that $$a_{n,m}$$ must separate variables and be periodic with period two with respect to each variable.

We can summarize the above analysis for the symmetries of equation (13) in the following statement which can be proven by direct computations.

Theorem 3.1.

Let function $$a_{n,m}$$ be such that $$a_{n+1,m+1} + a_{n,m} = a_{n+1,m}+a_{n,m+1}$$. Then in the $$n$$ direction, equation (13) admits

• 1.First order symmetries provided that $$a_{n+1,m}=a_{n,m}$$, which are generated by
\begin{align} \partial_{x} v_{n,m} = v_{n,m} \left({\cal{S}}-1\right) \frac{1}{v_{n-1,m} v_{n,m}-a_{n,m}},\quad \partial_{y} v_{n,m} = v_{n,m} \left({\cal{S}}-1\right) \frac{n-1}{v_{n-1,m} v_{n,m}-a_{n,m}} ,\quad \partial_{y} a_{n,m} =-1,\notag\\ \end{align}
(16)
where the $$y$$ flow is a master symmetry of the $$x$$ one.
• 2.Second order symmetries if $$a_{n+2,m}=a_{n,m}$$ and $$a_{n+1,m} \ne a_{n,m}$$, given by
$$\partial_{t} v_{n,m} = v_{n,m}^2 \left( \frac{1}{F_{(+)}} \frac{ v_{n+1,m}}{\omega_{n+1,m}}\,- \frac{1}{F_{(-)}} \frac{v_{n-1,m}}{\omega_{n-1,m}} + \, \frac{v_{n+1,m}-v_{n-1,m}}{F_{(-)} F_{(+)}}\,\frac{a_{n,m}-v_{n,m} (v_{n+1,m}+v_{n-1,m})}{\omega_{n,m}} \right)\!,$$
(17)
and
$$\partial_{s} v_{n,m} = (-1)^n \left(\partial_{t} v_{n,m}+\, \frac{2 v_{n,m}^2 (v_{n+1,m}-v_{n-1,m})}{(a_{n+1,m}-a_{n,m}) \omega_{n,m}}\right)\!,$$
(18)
where $$F_{(\pm)} = v_{n \pm 1} v_{n,m} - a_{n,m}$$ and $$\omega_{n,m} = v_{n+1,m} v_{n,m}^2 v_{n-1,m} - a_{n+1,m} v_{n,m} (v_{n+1,m}+v_{n-1,m}) + a_{n,m} a_{n+1,m}$$.
• 3.If $$a_{n,m}$$ is periodic in $$n$$ with period two and independent of $$m$$, i.e. $$a_{n+2,m}=a_{n,m}$$, $$a_{n+1,m} \ne a_{n,m}$$ and $$a_{n,m+1}=a_{n,m}$$, then there is one more second order symmetry, namely
\begin{eqnarray} \partial_{\tau} v_{n,m} &=& a_{n,m} (a_{n,m} - a_{n+1,m}) \left(n \, \partial_{t} v_{n,m}+\, v_{n,m}^2 \left(\frac{v_{n+1,m}}{F_{(+)} \omega_{n+1,m}} + \frac{v_{n-1,m}}{F_{(-)} \omega_{n-1,m}} \right) \right)\nonumber \\ & & - \,\frac{n \, (a_{n,m}+a_{n+1,m}) v_{n,m}^2 (v_{n+1,m}-v_{n-1,m})}{\omega_{n,m}} + \frac{v_{n,m} (v_{n+1,m} v_{n,m}^2 v_{n-1,m}-a_{n,m}^2)}{F_{(+)} F_{(-)}}. \end{eqnarray}
(19)

In the $$m$$ direction, the lowest order symmetries of equation (13) are of order one if $$a_{n,m+1} = a_{n,m}$$, and of order two if $$a_{n,m+2}=a_{n,m}$$ and $$a_{n,m+1} \ne a_{n,m}$$. The formulae for these symmetries follow from (16) to (19) by applying the changes $$(v_{n+i,m},a_{n+i,m},n,{\cal{S}}) \rightarrow (v_{n,m+i},-a_{n,m+i},m,{\cal{T}})$$.

When $$a_{n+2,m} = a_{n,m}$$, equation (13) and its symmetries are related to the equations in the previous section by the Miura transformation
$$\frac{z_{n,m}+1}{a_{n+1,m}- a_{n,m}} = \frac{v_{n,m} v_{n+1,m}}{v_{n+2,m} v_{n+1,m}^2 v_{n,m} - a_{n,m} v_{n+1,m} (v_{n+2,m}+v_{n,m}) + a_{n,m} a_{n+1,m}}.$$
(20)

Specifically it maps equation (13) to $$W_{n,m}(a_{n+1,m},a_{n,m},0)=0$$ in (6), and symmetries (17) and (18) to (9) with $$c_n=1$$ and $$c_n =(-1)^n$$, respectively.

Finally, at the limit $$a_{n,m} \rightarrow \alpha$$, equation (13) reduces to (15) and in the same way (17)–(19) reduce to symmetries of the latter equation. Specifically, (19) reduces to the $$n$$ dependent symmetry in (16), (17) becomes the second order symmetry of (15) and (18) after multiplying it with the constant $$(a_{n,m}-a_{n+1,m}) (-1)^n$$ leads to the first order autonomous symmetry in (16).

### 3.2 The discrete potential KdV equation

The next family of equations, we are going to consider is
$$(u_{n,m}-u_{n+1,m+1}) (u_{n+1,m}-u_{n,m+1}) - \alpha_n + \beta_m =0.$$
(21)
This is a generic choice since equation $$(u_{n,m}-u_{n+1,m+1}) (u_{n+1,m}-u_{n,m+1}) - f_{n,m} =0$$ and the symmetry analysis imply that $$f_{n,m}$$ must separate variables, i.e. $$f_{n,m} = \alpha_{n}-\beta_{m}$$. Moreover, the covariance of the equation allows us to perform the symmetry analysis in one direction and then apply changes $$\left(u_{n+i,m+j},\alpha_{n+i},\beta_{m+i},n,m\right) \rightarrow \left(u_{n+j,m+i},\beta_{m+i},\alpha_{n+i},m,n\right)$$ to derive the results in the other direction. Herein, we present the analysis for the first direction using condition (2) which now becomes
$${\cal{T}}(R) = \frac{u_{n-1,m}-u_{n,m+1}}{u_{n-1,m+1}-u_{n,m}}\, \frac{u_{n+N-1,m+1}-u_{n+N,m}}{u_{n+N,m+1}-u_{n+N-1,m}} \,R,\quad R = R(n,m,u_{n-N,m},\ldots,u_{n+N,m}).$$
(22)

Assuming that equation (21) admits a first order formal recursion operator, condition (22) with $$N=1$$ leads to a system of partial differential equations which is consistent provided that either $$R=0$$ or $$\alpha_{n+1} = \alpha_n$$. Hence equation (21) admits symmetries of order one only if $$\alpha_{n+1}=\alpha_{n}$$ and they can be found in [23]. To find the equations in family (21) admitting symmetries of order two but not of order one, we consider (22) with $$N=2$$, along with $$\alpha_{n+1} \ne \alpha_n$$ to exclude symmetries of lower order. The analysis of this equation implies that $$\alpha_n$$ must be a function of period two, $$\alpha_{n+2} = \alpha_{n}$$, and for the corresponding symmetries we can state the following result which can be shown by direct computations.

Theorem 3.2
The lowest order symmetries of equation (21) with $$\alpha_{n+2}=\alpha_{n}$$ and $$\alpha_{n+1}\ne \alpha_{n}$$ are generated by
$$\partial_{t} u_{n,m} = \frac{1}{u_{n-1,m}-u_{n+1,m}} \left(\frac{g_n}{h_{n,m}} + \frac{g_{n-1}}{h_{n-1,m}}\right) + \frac{g_{n} - g_{n-1}}{\alpha_{n}-\alpha_{n+1}} \,\frac{1}{u_{n-1,m}-u_{n+1,m}}, \qquad g_{n+2} = g_{n},$$
(23)
where $$h_{n,m} = (u_{n,m}-u_{n+2,m}) (u_{n+1,m}-u_{n-1,m}) - \alpha_n + \alpha_{n+1}$$. Moreover, the Miura transformation
$$\frac{z_{n,m}+1}{\alpha_{n+1}-\alpha_n} = \frac{1}{ (u_{n,m}-u_{n+2,m}) (u_{n+1,m}-u_{n-1,m}) + \alpha_{n+1} - \alpha_n}$$
(24)
maps (21) to $$W_{n,m}(\alpha_{n},\alpha_{n+1},\beta_m)=0$$ in (6), and symmetry (23) to (9) with $$c_n= g_n/(\alpha_{n}-\alpha_{n+1})^2$$.

Symmetries (23), as well as (28) below, were first given in an equivalent form in [14].

### 3.3 Deautonomization of the ABS equations

We can go through the same analysis starting with any of the ABS equations H1–H3, Q1–Q31 [6] or the deformed H equations dH1–dH3 [22] and derive corresponding integrable equations depending on a periodic function of $$n$$. We summarize these results in the following statement which can be verified by direct computations.

Theorem 3.3
Consider the equation
$$Q(u_{n,m+1},u_{n,m},u_{n+1,m},u_{n+1,m+1};\alpha_n,\beta_m) =0,$$
(25)
where $$Q(u_{n,m+1},u_{n,m},u_{n+1,m},u_{n+1,m+1};\alpha,\beta)$$ is the defining polynomial of any of the equations H1–H3, Q1–Q3 or dH1–dH3.

1. 1. If $$\alpha_{n}$$ is constant, $$\alpha_{n+1}=\alpha_{n}$$, then equation (25) admits first order symmetries in the $$n$$ direction [22, 23]

2. 2. If $$\alpha_{n+2} = \alpha_n$$ and $$\alpha_{n+1} \ne \alpha_n$$, then the lowest order symmetries of (25) are generated by
\begin{align} \partial_{t} u_{n,m} = \frac{f_{n-1,m} f_{n,m}}{\left(\partial_{u_{n+2,m}} h_{n,m} \right)} \left(\frac{g_n}{h_{n,m}} + \frac{g_{n-1}}{h_{n-1,m}}\right) + \frac{g_{n} - g_{n-1}}{k(\alpha_{n},\alpha_{n+1})} \,\frac{P_{n,m}}{\left(\partial_{u_{n+2,m}} h_{n,m} \right)}, \qquad {\mbox{with}}\quad g_{n+2} = g_{n},\notag\\ \end{align}
(26a)
where
$$\begin{array}{l} h_{n,m} = Q(u_{n-1,m},u_{n,m},u_{n+1,m},u_{n+2,m};\alpha_n,\alpha_{n+1}),\\ \\ f_{n,m} = f(n,m,u_{n,m},u_{n+1,m},\alpha_{n}) = \frac{\left(\partial_{u_{n+2,m}} h_{n,m}\right) \left(\partial_{u_{n-1,m}} h_{n,m}\right)- h_{n,m} \left(\partial^2_{u_{n+2,m},u_{n-1,m}}h_{n,m}\right)}{k(\alpha_{n},\alpha_{n+1})}, \\ \\ k(\alpha_{n},\alpha_{n+1}) = \left\{ \begin{array}{ll} \alpha_{n}-\alpha_{n+1}, & {\mbox{for H1, H2, dH1, dH2}} \\ \alpha_{n+1}^2-\alpha_{n}^2 , & {\mbox{for H3, dH3}} \\ \alpha_{n} \alpha_{n+1} (\alpha_{n}-\alpha_{n+1}), & {\mbox{for Q1, Q2}} \\ (\alpha_{n}^2-\alpha_{n+1}^2) (\alpha_{n}^2-1) ( \alpha_{n+1}^2-1), & {\mbox{for Q3}} \end{array} \right. \end{array}$$
(26b)
and functions $$P_{n,m} = P(n,m,u_{n-1,m},u_{n,m},u_{n+1,m})$$ are given in the following table.

Equation $$P_{n,m}$$
H1 $$1$$
dH1 $$1 - \epsilon \left( u_{n-1,m} u_{n+1,m} {\cal{X}}_{n,m} +u_{n,m}^2 {\cal{Y}}_{n,m}\right)$$
H2 $$u_{n-1,m} + 2 u_{n,m} + u_{n+1,m} + \alpha_{n} + \alpha_{n+1}$$
dH2 $$u_{n-1,m} + 2 u_{n,m} + u_{n+1,m} + \alpha_{n} + \alpha_{n+1} - 4 \epsilon \left( (\alpha_{n} u_{n-1,m} + \alpha_{n+1} u_{n+1,m}) {\cal{X}}_{n,m} + {\phantom{\tfrac{\alpha_{n}^2}{2}}} \right.$$
$$\left. \quad \qquad (\alpha_n \alpha_{n+1} + u_{n-1,m} u_{n+1,m}) {\cal{X}}_{n,m}^2 + \left( \tfrac{\alpha_{n}^2 + \alpha_{n+1}^2}{2} + (\alpha_{n}+ \alpha_{n+1}) u_{n,m} + u_{n,m}^2\right) {\cal{Y}}_{n,m}^2\right)$$
H3 $$\tfrac{1}{2} \left( \alpha_{n} u_{n-1,m} u_{n,m} + \alpha_{n+1} u_{n,m} u_{n+1,m} + 2 \delta \alpha_{n} \alpha_{n+1}\right)$$
dH3 $$\tfrac{1}{2} \left( \alpha_{n} u_{n-1,m} u_{n,m} + \alpha_{n+1} u_{n,m} u_{n+1,m} + 2 \delta \alpha_{n} \alpha_{n+1}\right) - \epsilon \left(u_{n-1,m} u_{n+1,m} {\cal{X}}_{n,m} + \tfrac{\alpha_{n}^2 + \alpha_{n+1}^2}{2 \alpha_{n} \alpha_{n+1}} u_{n,m}^2 {\cal{Y}}_{n,m}\right)$$
Q1 $$(u_{n,m}-u_{n-1,m}) (u_{n,m}-u_{n+1,m}) - \delta^2 \alpha_{n} \alpha_{n+1}$$
Q2 $$(u_{n,m}-u_{n-1,m}) (u_{n,m}-u_{n+1,m}) - \alpha_{n+1}^2 u_{n+1,m} + (\alpha_{n}^2 + \alpha_{n+1}^2 -4 \alpha_{n} \alpha_{n+1}) u_{n,m} - \alpha_{n}^2 u_{n-1,m} + \alpha_{n}^2 \alpha_{n+1}^2$$
Q3 $$\alpha_{n} \alpha_{n+1} u_{n-1,m} u_{n+1,m} - \left(\tfrac{\alpha_{n} (\alpha_{n+1}^2+1)}{2} u_{n+1,m} + \tfrac{\alpha_{n+1} (\alpha_{n}^2+1)}{2} u_{n-1,m} \right) u_{n,m} + \tfrac{\alpha_{n}^2 +\alpha_{n+1}^2}{2} u_{n,m}^2 + \tfrac{\delta^2 (\alpha_{n}^2-1) (\alpha_{n+1}^2-1)}{4}$$
Equation $$P_{n,m}$$
H1 $$1$$
dH1 $$1 - \epsilon \left( u_{n-1,m} u_{n+1,m} {\cal{X}}_{n,m} +u_{n,m}^2 {\cal{Y}}_{n,m}\right)$$
H2 $$u_{n-1,m} + 2 u_{n,m} + u_{n+1,m} + \alpha_{n} + \alpha_{n+1}$$
dH2 $$u_{n-1,m} + 2 u_{n,m} + u_{n+1,m} + \alpha_{n} + \alpha_{n+1} - 4 \epsilon \left( (\alpha_{n} u_{n-1,m} + \alpha_{n+1} u_{n+1,m}) {\cal{X}}_{n,m} + {\phantom{\tfrac{\alpha_{n}^2}{2}}} \right.$$
$$\left. \quad \qquad (\alpha_n \alpha_{n+1} + u_{n-1,m} u_{n+1,m}) {\cal{X}}_{n,m}^2 + \left( \tfrac{\alpha_{n}^2 + \alpha_{n+1}^2}{2} + (\alpha_{n}+ \alpha_{n+1}) u_{n,m} + u_{n,m}^2\right) {\cal{Y}}_{n,m}^2\right)$$
H3 $$\tfrac{1}{2} \left( \alpha_{n} u_{n-1,m} u_{n,m} + \alpha_{n+1} u_{n,m} u_{n+1,m} + 2 \delta \alpha_{n} \alpha_{n+1}\right)$$
dH3 $$\tfrac{1}{2} \left( \alpha_{n} u_{n-1,m} u_{n,m} + \alpha_{n+1} u_{n,m} u_{n+1,m} + 2 \delta \alpha_{n} \alpha_{n+1}\right) - \epsilon \left(u_{n-1,m} u_{n+1,m} {\cal{X}}_{n,m} + \tfrac{\alpha_{n}^2 + \alpha_{n+1}^2}{2 \alpha_{n} \alpha_{n+1}} u_{n,m}^2 {\cal{Y}}_{n,m}\right)$$
Q1 $$(u_{n,m}-u_{n-1,m}) (u_{n,m}-u_{n+1,m}) - \delta^2 \alpha_{n} \alpha_{n+1}$$
Q2 $$(u_{n,m}-u_{n-1,m}) (u_{n,m}-u_{n+1,m}) - \alpha_{n+1}^2 u_{n+1,m} + (\alpha_{n}^2 + \alpha_{n+1}^2 -4 \alpha_{n} \alpha_{n+1}) u_{n,m} - \alpha_{n}^2 u_{n-1,m} + \alpha_{n}^2 \alpha_{n+1}^2$$
Q3 $$\alpha_{n} \alpha_{n+1} u_{n-1,m} u_{n+1,m} - \left(\tfrac{\alpha_{n} (\alpha_{n+1}^2+1)}{2} u_{n+1,m} + \tfrac{\alpha_{n+1} (\alpha_{n}^2+1)}{2} u_{n-1,m} \right) u_{n,m} + \tfrac{\alpha_{n}^2 +\alpha_{n+1}^2}{2} u_{n,m}^2 + \tfrac{\delta^2 (\alpha_{n}^2-1) (\alpha_{n+1}^2-1)}{4}$$
Equation $$P_{n,m}$$
H1 $$1$$
dH1 $$1 - \epsilon \left( u_{n-1,m} u_{n+1,m} {\cal{X}}_{n,m} +u_{n,m}^2 {\cal{Y}}_{n,m}\right)$$
H2 $$u_{n-1,m} + 2 u_{n,m} + u_{n+1,m} + \alpha_{n} + \alpha_{n+1}$$
dH2 $$u_{n-1,m} + 2 u_{n,m} + u_{n+1,m} + \alpha_{n} + \alpha_{n+1} - 4 \epsilon \left( (\alpha_{n} u_{n-1,m} + \alpha_{n+1} u_{n+1,m}) {\cal{X}}_{n,m} + {\phantom{\tfrac{\alpha_{n}^2}{2}}} \right.$$
$$\left. \quad \qquad (\alpha_n \alpha_{n+1} + u_{n-1,m} u_{n+1,m}) {\cal{X}}_{n,m}^2 + \left( \tfrac{\alpha_{n}^2 + \alpha_{n+1}^2}{2} + (\alpha_{n}+ \alpha_{n+1}) u_{n,m} + u_{n,m}^2\right) {\cal{Y}}_{n,m}^2\right)$$
H3 $$\tfrac{1}{2} \left( \alpha_{n} u_{n-1,m} u_{n,m} + \alpha_{n+1} u_{n,m} u_{n+1,m} + 2 \delta \alpha_{n} \alpha_{n+1}\right)$$
dH3 $$\tfrac{1}{2} \left( \alpha_{n} u_{n-1,m} u_{n,m} + \alpha_{n+1} u_{n,m} u_{n+1,m} + 2 \delta \alpha_{n} \alpha_{n+1}\right) - \epsilon \left(u_{n-1,m} u_{n+1,m} {\cal{X}}_{n,m} + \tfrac{\alpha_{n}^2 + \alpha_{n+1}^2}{2 \alpha_{n} \alpha_{n+1}} u_{n,m}^2 {\cal{Y}}_{n,m}\right)$$
Q1 $$(u_{n,m}-u_{n-1,m}) (u_{n,m}-u_{n+1,m}) - \delta^2 \alpha_{n} \alpha_{n+1}$$
Q2 $$(u_{n,m}-u_{n-1,m}) (u_{n,m}-u_{n+1,m}) - \alpha_{n+1}^2 u_{n+1,m} + (\alpha_{n}^2 + \alpha_{n+1}^2 -4 \alpha_{n} \alpha_{n+1}) u_{n,m} - \alpha_{n}^2 u_{n-1,m} + \alpha_{n}^2 \alpha_{n+1}^2$$
Q3 $$\alpha_{n} \alpha_{n+1} u_{n-1,m} u_{n+1,m} - \left(\tfrac{\alpha_{n} (\alpha_{n+1}^2+1)}{2} u_{n+1,m} + \tfrac{\alpha_{n+1} (\alpha_{n}^2+1)}{2} u_{n-1,m} \right) u_{n,m} + \tfrac{\alpha_{n}^2 +\alpha_{n+1}^2}{2} u_{n,m}^2 + \tfrac{\delta^2 (\alpha_{n}^2-1) (\alpha_{n+1}^2-1)}{4}$$
Equation $$P_{n,m}$$
H1 $$1$$
dH1 $$1 - \epsilon \left( u_{n-1,m} u_{n+1,m} {\cal{X}}_{n,m} +u_{n,m}^2 {\cal{Y}}_{n,m}\right)$$
H2 $$u_{n-1,m} + 2 u_{n,m} + u_{n+1,m} + \alpha_{n} + \alpha_{n+1}$$
dH2 $$u_{n-1,m} + 2 u_{n,m} + u_{n+1,m} + \alpha_{n} + \alpha_{n+1} - 4 \epsilon \left( (\alpha_{n} u_{n-1,m} + \alpha_{n+1} u_{n+1,m}) {\cal{X}}_{n,m} + {\phantom{\tfrac{\alpha_{n}^2}{2}}} \right.$$
$$\left. \quad \qquad (\alpha_n \alpha_{n+1} + u_{n-1,m} u_{n+1,m}) {\cal{X}}_{n,m}^2 + \left( \tfrac{\alpha_{n}^2 + \alpha_{n+1}^2}{2} + (\alpha_{n}+ \alpha_{n+1}) u_{n,m} + u_{n,m}^2\right) {\cal{Y}}_{n,m}^2\right)$$
H3 $$\tfrac{1}{2} \left( \alpha_{n} u_{n-1,m} u_{n,m} + \alpha_{n+1} u_{n,m} u_{n+1,m} + 2 \delta \alpha_{n} \alpha_{n+1}\right)$$
dH3 $$\tfrac{1}{2} \left( \alpha_{n} u_{n-1,m} u_{n,m} + \alpha_{n+1} u_{n,m} u_{n+1,m} + 2 \delta \alpha_{n} \alpha_{n+1}\right) - \epsilon \left(u_{n-1,m} u_{n+1,m} {\cal{X}}_{n,m} + \tfrac{\alpha_{n}^2 + \alpha_{n+1}^2}{2 \alpha_{n} \alpha_{n+1}} u_{n,m}^2 {\cal{Y}}_{n,m}\right)$$
Q1 $$(u_{n,m}-u_{n-1,m}) (u_{n,m}-u_{n+1,m}) - \delta^2 \alpha_{n} \alpha_{n+1}$$
Q2 $$(u_{n,m}-u_{n-1,m}) (u_{n,m}-u_{n+1,m}) - \alpha_{n+1}^2 u_{n+1,m} + (\alpha_{n}^2 + \alpha_{n+1}^2 -4 \alpha_{n} \alpha_{n+1}) u_{n,m} - \alpha_{n}^2 u_{n-1,m} + \alpha_{n}^2 \alpha_{n+1}^2$$
Q3 $$\alpha_{n} \alpha_{n+1} u_{n-1,m} u_{n+1,m} - \left(\tfrac{\alpha_{n} (\alpha_{n+1}^2+1)}{2} u_{n+1,m} + \tfrac{\alpha_{n+1} (\alpha_{n}^2+1)}{2} u_{n-1,m} \right) u_{n,m} + \tfrac{\alpha_{n}^2 +\alpha_{n+1}^2}{2} u_{n,m}^2 + \tfrac{\delta^2 (\alpha_{n}^2-1) (\alpha_{n+1}^2-1)}{4}$$

3. 3.For the non-autonomous ABS equations, the Miura transformation
$$\frac{p_n}{k(\alpha_{n+1},\alpha_{n}) }\, \left( z_{n,m}+1\right) \,= \,\frac{ f_{n,m}}{h_{n,m}}, \quad {\mbox{where }} \, p_n \,=\, \left\{ \begin{array}{ll} 1, & {\mbox{for H1, H2}} \\ \alpha_{n}, & {\mbox{for H3, Q1, Q2}} \\ \alpha_{n+1} (\alpha_{n}^2-1), & {\mbox{for Q3}} \end{array} \right.$$
(27)
maps the difference equation (25) to the quadratic non-autonomous quad equation (6) with parameters
$$(A_n,B_m)\,=\, \left\{ \begin{array}{ll} (\alpha_{n},\beta_{m}), & {\mbox{for H1, H2}} \\ ( \beta_m^2 \alpha_n^2, \beta_m^2 (\alpha_{n+1}^2 +\alpha_n^2) - \alpha_n^2 \alpha_{n+1}^2), & {\mbox{for H3}} \\ (\beta_m \alpha_n, \beta_m (\alpha_{n+1} +\alpha_n) - \alpha_n \alpha_{n+1}), & {\mbox{for Q1, Q2}} \\ (\beta_m^2 \alpha_n^2 + \alpha_{n+1}^2, (\alpha_{n+1}^2 +\alpha_n^2 - \alpha_n^2 \alpha_{n+1}^2) + \beta_m^2 (\alpha_n^2+ \alpha_{n+1}^2-1)), &{\mbox{for Q3}} \end{array} \right.$$
and its symmetry (26) to (9) with $$c_n = g_n p_n^2/k(\alpha_{n+1},\alpha_{n})^2$$.
4. 4.If $$\alpha_{n+2} = \alpha_n$$ with $$\alpha_{n+1} \ne \alpha_n$$, and $$\beta_m = \beta\in {\mathbb{R}}$$, then equations H1 and H2 admit one more symmetry, namely
\begin{eqnarray} {\mbox{H1:}} && \partial_{\tau} u_{n,m} = \partial_{t} u_{n,m} - u_{n,m}, \quad {\mbox{with }}\,\, g_{n} = n (\alpha_{n+1} - \beta) (\alpha_{n}-\alpha_{n+1}), \\ \end{eqnarray}
(28)

\begin{eqnarray} {\mbox{H2:}} && \partial_{\tau} u_{n,m} = \partial_{t} u_{n,m} + \frac{2 u_{n,m} + \beta}{(\beta -\alpha_{n}) (\beta-\alpha_{n+1})}, \quad {\mbox{with }}\,\,\, g_{n} = \frac{n (\alpha_{n}-\alpha_{n+1})}{\beta-\alpha_{n}}. \end{eqnarray}
(29)

In the $$m$$ direction, the lowest order symmetries of equation (25) are of order one if $$\beta_{m+1} = \beta_{m}$$, and of order two if $$\beta_{m+2}=\beta_{m}$$ and $$\beta_{m+1} \ne \beta_{m}$$. The formulae for latter symmetries follow from (26), (28) and (29) by applying the changes $$(u_{n+i,m},\alpha_{n+i}) \rightarrow (u_{n,m+i},\beta_{m+i})$$.

At the limit $$\alpha_n \rightarrow \alpha$$, equation (25) reduces to the original autonomous equation and (26) yields the first two symmetries of the latter equation. Indeed, symmetry (26) with $$g_n=1$$ reduces to the second order symmetry of the corresponding autonomous equation [4], whereas with $$g_n$$ such that $$g_n-g_{n+1}= k(\alpha_{n},\alpha_{n+1})$$ it reduces to a linear combination of the first and the second order symmetry of the original equation. Finally, at the limit $$\alpha_n \rightarrow \alpha$$, (28) and (29) lead to the $$n$$ dependent first order symmetries of the autonomous equations H1 and H2, respectively, given in [23].

Another interesting characteristic of (26) with $$g_n=1$$ is that it can be interpreted as a non-autonomous extension of
$$\partial_{t} u_{n,m} = \frac{f(u_{n-1,m},u_{n,m}) f(u_{n,m},u_{n+1,m}) (u_{n+2,m}-u_{n-2,m})}{h(u_{n-1,m},u_{n,m},u_{n+1,m},u_{n+2,m}) h(u_{n-2,m},u_{n-1,m},u_{n,m},u_{n+1,m})},$$
(30)
where $$h$$ is an affine linear polynomial possessing the symmetries of the square and $$f(u_{n,m},u_{n+1,m})$$ is determined by $$h(u_{n-1,m},u_{n,m},u_{n+1,m},u_{n+2,m})$$ in the similar way $$f_{n,m}$$ is determined from $$h_{n,m}$$ in (26b). These equations were studied recently in [17] in connection with seven point difference equations and Q4, whereas their relation to equation (9) with $$c_n=1$$ is given by the Miura transformation (27) with $$\tfrac{p_n}{k(\alpha_{n+1},\alpha_{n})}= 1$$. The connection of $$h$$ in (30) with the defining polynomial of Q4 as described in [17] as a limiting process differs from relation (26b). We can think the latter as straightening an elementary quadrilateral by identifying parameter $$\beta$$ with $$\alpha_{n-1}=\alpha_{n+1}$$ and $$u_{n,m+1}$$, $$u_{n+1,m+1}$$ with $$u_{n-1,m}$$ and $$u_{n+2,m}$$, respectively.

### 3.4 Other deautonomized partial difference equations

We can apply the same procedure to other equations and derive their non-autonomous integrable counterparts which admit higher order symmetries. Herein, we present the discrete sine-Gordon and non-autonomous Hietarinta–Viallet equations, as well as the deautonomization of H1 admitting symmetries of order three in the $$n$$ direction.

#### 3.4.1 Discrete sine-Gordon equation

The deautonomized discrete sine-Gordon equation
$$\alpha_n (u_{n,m} u_{n+1,m+1} - u_{n+1,m} u_{n,m+1}) - \beta_m (u_{n,m} u_{n+1,m} u_{n,m+1} u_{n+1,m+1}-1) = 0,$$
(31)
where $$\alpha_{n}$$ and $$\beta_{m}$$ are non-constant periodic functions with period two, possesses generalized symmetries in both directions the lowest order of which is two. The first symmetry in the $$n$$ direction is generated by
$$\partial_{t} u_{n,m} = \frac{u_{n-1,m} u_{n,m}^2 u_{n+1,m} (u_{n+2,m}-u_{n-2,m})}{\tilde{h}_{n,m} \tilde{h}_{n-1,m}}$$
(32)
where $$\tilde{h}_{n,m} = \alpha_{n+1} (u_{n+2,m} u_{n-1,m} + u_{n+1,m} u_{n,m}) + \alpha_{n} (u_{n+2,m} u_{n+1,m} + u_{n,m} u_{n-1,m})$$, and the corresponding one in the other direction follows from (32) by changing $$u_{n+i,m}$$ to $$u_{n,m+i}$$ and $$\alpha_{n+i}$$ to $$\beta_{m+1+i}$$. Up to a point transformation, differential-difference equation (32) is the symmetry of the deautonomized H3 with $$\delta=0$$.

#### 3.4.2 Hietarinta–Viallet equation

Equation
$$\alpha_{n} w_{n,m} w_{n,m+1} - w_{n,m+1} w_{n+1,m+1} + w_{n+1,m} w_{n+1,m+1}= 0,\quad \alpha_{n+2} = \alpha_n,\,\, \alpha_n \ne \alpha_{n+1},$$
(33)
is a deautonomization of $$w_{n,m} w_{n,m+1} - w_{n,m+1} w_{n+1,m+1} + w_{n+1,m} w_{n+1,m+1}=0$$, an equation due to Hietarinta and Viallet [24]. Its lowest order generalized symmetries in the $$n$$ direction are of order two generated by
\begin{eqnarray} \partial_{t} w_{n,m} &=& \frac{w_{n,m}}{(\partial_{w_{n+2,m}} \chi_{n,m})} \left(w_{n-1,m}w_{n,m} w_{n+1,m} \left(\frac{1}{\chi_{n,m}} + \frac{1}{\alpha_{n} \chi_{n-1,m}}\right)- \frac{w_{n+1,m}}{\alpha_n \alpha_{n+1}}\right)\!, \\ \end{eqnarray}
(34a)

\begin{eqnarray} \partial_{s} w_{n,m} &=& \frac{(-1)^n w_{n,m}}{(\partial_{w_{n+2,m}} \chi_{n,m})} \left(w_{n-1,m}w_{n,m} w_{n+1,m} \left(\frac{1}{\chi_{n,m}} - \frac{1}{\alpha_{n} \chi_{n-1,m}}\right)- \frac{\alpha_n +\alpha_{n+1}}{\alpha_n-\alpha_{n+1}} \frac{ w_{n+1,m}}{\alpha_n \alpha_{n+1}}\right)\!,\\ \end{eqnarray}
(34b)

\begin{eqnarray} {\mbox{and}}\quad \partial_\tau w_{n,m} &=& \frac{n \,\alpha_{n+1} \,w_{n,m}}{(\partial_{w_{n+2,m}} \chi_{n,m})} \left( (\alpha_{n}-\alpha_{n+1}) w_{n-1,m} w_{n,m} w_{n+1,m} \left(\frac{1}{\chi_{n,m}}- \frac{1}{\alpha_{n-1} \chi_{n-1,m}}\right) - 2 w_{n-1,m}\right) \nonumber \\ && +\, \frac{w_{n,m} w_{n+1,m} (w_{n,m}-\alpha_{n} w_{n-2,m})}{\chi_{n-1,m}},\qquad {\mbox{along with }} \quad \partial_\tau \alpha_{n} = 2\,\alpha_{n}\,, \end{eqnarray}
(34c)
where $$\chi_{n,m} := w_{n+2,m} ( w_{n+1,m} - \alpha_{n+1} w_{n-1,m}) - \alpha_{n+1} w_{n,m} (w_{n+1,m}-\alpha_{n} w_{n-1,m})$$, whereas the symmetries in the $$m$$ direction are generated by
$$\partial_{y} w_{n,m} = \frac{w_{n,m} w_{n,m-1}}{w_{n,m+1}}.$$
Moreover, the Miura transformation
$$\frac{1+v_{n,m}}{\alpha_n -\alpha_{n+1}} = \frac{w_{n,m} w_{n+1,m}}{w_{n+2,m} ( w_{n+1,m} - \alpha_{n+1} w_{n-1,m}) - \alpha_{n+1} w_{n,m} (w_{n+1,m}-\alpha_{n} w_{n-1,m})}$$
maps equation (33) to $$W_{n,m}(\alpha_n,\alpha_{n+1},0)=0$$ and its symmetries (34a) and (34b) to equation (9) with $$c_n = 1/(\alpha_{n}-\alpha_{n+1})^2$$ and $$c_n = (-1)^n/(\alpha_{n}-\alpha_{n+1})^2$$, respectively.

#### 3.4.3 Discrete potential KdV equation and third order symmetries

In all our previous examples, we have considered equations admitting symmetries of order one or two. We can continue and consider deautonomized equations admitting symmetries of order three in the $$n$$ direction. But this is a very difficult computational task as the calculations are lengthy and cumbersome. We were able to partially analyse the discrete potential KdV equation, and we can present the outcome of our analysis in the following form.

Theorem 3.4
The lowest order generalized symmetries in the $$n$$ direction of equation
$$(u_{n,m}-u_{n+1,m+1}) (u_{n+1,m}-u_{n,m+1}) - \alpha_n + \beta =0,\quad \alpha_{n+3} = \alpha_n, \quad \alpha_{n+2} \ne \alpha_{n},$$
(35)
are of order three. A symmetry is generated by
\begin{eqnarray} \partial_{t^3} u_{n,m} &=& \frac{g_n}{h_{(+)} H_{n+1,m}} - \frac{g_{n-2}}{h_{(-)} H_{n-1,m}} + \frac{g_{n-1} (u_{n+2,m}-u_{n-2,m})^2}{h_{(+)} h_{(-)} H_{n,m}} + \frac{g_{n-1}-g_{n}}{(\alpha_{n+1}-\alpha_{n+2}) w_{n,m} h_{(+)}} \nonumber \\ & & + \frac{g_{n-2}-g_{n-1}}{(\alpha_{n+1}-\alpha_{n}) w_{n,m} h_{(-)}} + \frac{\alpha_{n} (g_n-g_{n-1}) + \alpha_{n+1} (g_{n-2} -g_n) + \alpha_{n+2} (g_{n-1}-g_{n-2})}{(\alpha_{n}-\alpha_{n+1}) (\alpha_{n+1}-\alpha_{n+2}) (\alpha_{n+2}-\alpha_{n}) w_{n,m}}, \end{eqnarray}
(36a)
where $$g_{n+3}= g_n$$ is an arbitrary function, $$w_{n,m} = u_{n+1,m}-u_{n-1,m}$$ and
\begin{eqnarray} h_{(\pm)} &=& (u_{n,m}-u_{n \pm 2,m}) (u_{n+1,m}-u_{n-1,m}) + \alpha_{n+2}-\alpha_{n},\\ \end{eqnarray}
(36b)

\begin{eqnarray} H_{n,m} &=& (u_{n+2,m}-u_{n,m}) (u_{n+1,m}-u_{n-1,m}) (u_{n,m} -u_{n-2,m}) + \nonumber \\ && \qquad \alpha_{n} (u_{n,m}-u_{n-2,m}) -\alpha_{n+1} (u_{n+2,m}-u_{n-2,m}) + \alpha_{n+2} (u_{n+2,m}-u_{n,m}). \end{eqnarray}
(36c)
Another symmetry is generated by
$$\partial_{\tau^3} u_{n,m} = \partial_{t^3} u_{n,m} - u_{n,m} \quad {\mbox{with}} \quad g_{n} = \frac{2}{3} n (\alpha_{n+2}-\beta) (\alpha_{n}-\alpha_{n+2}) (\alpha_{n+1}-\alpha_{n+2}).$$
(37)

Symmetry (36) yields the first three generalized symmetries of H1 at the limit $$\alpha_{n} \rightarrow \alpha$$. In particular, (36) with $$g_n=1$$ reduces to third order symmetry of H1, with $$g_n= \alpha_{n+2}$$ leads to a linear combination of the third and the second order symmetries, and with $$g_{n}=\alpha_{n} \alpha_{n+1}$$ yields a linear combination of the first three symmetries of H1. On the other hand, (37) reduces to the first order $$n$$ dependent symmetry of the original potential KdV equation [23]. It should be noted that equation (35) was also discussed in [14] and a third order symmetry was given there (with some misprints) which corresponds to (36) with $$g_n = \alpha_{n+1} (\alpha_{n+1}-\alpha_{n+2}) (\alpha_{n}-\alpha_{n+2})$$.

## 4. Differential-difference equations and reductions

Exact reductions of a differential-difference equation lead to ordinary difference equations and in particular, as it was conjectured in [25], every autonomous difference equation obtained by an exact reduction of the differential-difference equation is an integrable mapping. In this section, we present exact reductions of the non-autonomous equations (9), (17), (23), and of their conserved forms, which lead to non-autonomous ordinary difference equations and corresponding first integrals. As some of these reductions lead to alternating QRT maps [18, 19], these examples suggest that in certain cases the QRT conjecture is also valid for non-autonomous equations.

Example 1
We start with the stationary solutions of (9) which satisfy the non-autonomous ordinary difference equation $$\partial_{t} z_n = 0$$ (here and in the following examples, we suppress the second index $$m$$ as it does not vary). We can readily integrate the resulting equation twice and then write it as
$$z_{n+1} z_{n-1} = \frac{z_{n} (b_n -c_n z_{n})}{c_{n+1} (z_{n}+1)},\quad c_{n+2}=c_n,\,\,b_{n+2} = b_n,$$
(38)
where $$b_n$$ is the constant of integration. Moreover this equation admits the first integral
$$I_n = (-1)^n \left( (c_n z_{n} -b_n) \left(1 + \frac{1}{z_{n-1}}\right) - (c_{n+1} z_{n-1} -b_{n+1}) \left(1 + \frac{1}{z_{n}}\right)\right),$$
(39)
which follows from the first canonical conservation (11) of (9) presented in Section 2.
Example 2
We consider equation (17) with $$a_{n,m} = \alpha_{n}$$ being periodic with period two. Seeking for solutions of the form $$v(n,t) = \exp(\mu (-1)^n t) v_n$$, we substitute this form into (17) and after one integration we find that $$v_n$$ is related to the solution $$f_n$$ of the alternating QRT map
$$(f_{n+1}-f_{n}) (f_{n}-f_{n-1}) = \frac{(\alpha_{n}-\alpha_{n+1}) f_{n} (f_{n} + b_n) (f_{n}+b_{n+1})}{\alpha_{n+1} (f_{n}+\alpha_{n+1})},$$
(40)
a first integral of which is given by
$$I_n = \frac{\alpha_{n}^2 f_{n}^2 - \alpha_{n+1}^2 f_{n-1}^2 + (\alpha_{n} f_{n}- \alpha_{n+1} f_{n-1}-s_n) f_{n-1} f_{n}+ r_n f_{n} -r_{n-1} f_{n-1}}{f_{n}-f_{n-1}},$$
(41)
where $$s_n = \alpha_{n}^2-\alpha_{n+1}^2 - (\alpha_{n}-\alpha_{n+1}) (b_n+b_{n+1})$$ and $$r_n =(2 \alpha_{n}^2 \alpha_{n+1}^2 + \alpha_{n} (\alpha_{n}-\alpha_{n+1}) b_n b_{n+1})/(\alpha_{n}+\alpha_{n+1})$$. Specifically, the relation between $$v_n$$ and $$f_n$$ is
$$\frac{\alpha_{n+1} v_{n} v_{n+1}}{v_{n+2} v_{n+1}^2 v_{n} - \alpha_{n} v_{n+1} (v_{n+2}+v_{n}) + \alpha_{n} \alpha_{n+1}} = \frac{f_{n}+\alpha_{n+1}}{\alpha_{n+1}-\alpha_{n}},\quad v_{n} v_{n+1} = \alpha_{n} \frac{f_{n}+b_n}{f_{n-1}+b_n},$$
where $$b_n= \alpha_{n} \left(1+ \left(\lambda - \mu (-1)^{n}/2\right)\,\alpha_{n+1} (\alpha_{n+1}-\alpha_{n})\right)$$ is periodic with period two and $$\lambda$$ is the constant of integration.
Remark 4.1

The first integral (41) is the reduction of the first canonical conservation law of (17) which can be derived from (11) after using the Miura transformation (20).

Example 3
We seek for solutions to the differential-difference equation (23) of the form $$u(n,t) = (\lambda + \mu (-1)^n) t + u_n$$. Substituting into the equation and setting $$(u_{n+1}-u_{n-1})(u_{n} -u_{n-2}) + \alpha_{n+1}-\alpha_{n} = \delta_n^{-1}$$, we find that $$\delta_{n}$$ must satisfy equation
$$\left(\delta_{n-1} + \delta_{n}\right)\left(\delta_{n} + \delta_{n+1}\right) = \frac{\lambda^2-\mu^2}{\delta_{n}} +(\lambda^2-\mu^2) \left(\alpha_{n}-\alpha_{n+1}\right).$$
(42)
A first integral of this map is given by
$$I_n = \frac{\delta_{n} \delta_{n-1} (\delta_{n}+\delta_{n-1}) + (\lambda^2 - \mu^2) (\alpha_{n} \delta_{n} + \alpha_{n+1} \delta_{n-1}) + \lambda^2-\mu^2}{\delta_{n}+\delta_{n-1}}.$$
(43)
On the other hand, searching for solutions to (23) of the form $$u(n,t) = \exp(\mu (-1)^n t) u_n$$, we introduce function $$y_n$$ as $$u_n u_{n+1} = (\alpha_{n}-\alpha_{n+1}) y_n$$ to find that it must satisfy the ordinary difference equation
$$y_{n+3} y_{n+1}-y_{n+2} y_{n} - \mu_n F_{n} F_{n+1} = 0,$$
(44)
where $$F_{n} := (y_{n+2}+y_{n+1}) (y_{n+1} + y_{n}) - y_{n+1}$$ and $$\mu_n := \mu (\alpha_{n}-\alpha_{n+1})^2 (-1)^n$$. Equation (44) admits a two-integral $$J_n$$ and a first integral $$I_n$$ given by
$$J_n = \frac{y_{n+1}^2}{F_{n}^2} - \mu_n \,\frac{y_{n+1}^2 - y_{n+2} y_{n}}{F_{n}} \quad {\mbox{ and }} \quad I_n =\mu_n^3 y_{n+1}^3 + \frac{\mu_n y_{n+1}}{(y_{n+1}+y_n) F_n} \left( \frac{X_n}{F_n} -\mu_n Y_n \right) - \frac{y_{n+1}}{F_n^3} (F_n^2-2 y_{n+1}^2),$$
(45a)
respectively, where
\begin{eqnarray} X_n &=& 4 y_{n+1}^3 + (6 y_n +1) y_{n+1}^2 - (F_n^2 -2 y_n^2 - F_n +y_n) y_{n+1} - (F_n+1) F_n y_n,\\ \end{eqnarray}
(45b)

\begin{eqnarray} Y_n &=& 2 y_{n+1}^3 + 2 (y_n+1) y_{n+1}^2 + (F_n+2 y_n^2) y_{n+1} - F_n y_n. \end{eqnarray}
(45c)
Remark 4.2

The integrals follow from the reduction of the corresponding conserved forms of (23) which can be derived from (11) and (12) by employing the Miura transformation (24).

## 5. Difference equations and periodic reductions

The difference equations we derived in the previous section can also be interpreted as defining certain group invariant solutions of the related partial difference equations. In this section, we consider some particular periodic reductions of the latter equations and show how they lead to families of non-autonomous ordinary difference equations. Since the periodic functions appearing in the equations are specific, our approach differs from similar considerations in [15, 26] where the reduction determines also the form of the arbitrary functions involved in the equations.

To be more precise, in this section, we discuss periodic reductions of equations
$$v_{n,m} + \frac{\alpha_{n}}{v_{n,m+1}} - \frac{\alpha_{n+1}}{v_{n+1,m}} - v_{n+1,m+1}= 0,\quad \alpha_{n+2} = \alpha_{n} ,\,\, \alpha_{n+1} \ne \alpha_{n},$$
(46)
and
$$(u_{n,m}-u_{n+1,m+1}) (u_{n+1,m}-u_{n,m+1}) - \alpha_n=0,\quad \alpha_{n+2} = \alpha_n,\,\, \alpha_{n+1} \ne \alpha_n.$$
(47)

The former is (13) with $$a_{n,m} = \alpha_{n}$$ and the latter is (21) with $$\beta_{m} =0$$ where in both cases $$\alpha_{n}$$ is a non-constant and periodic with period two, i.e. $$\alpha_{n} = a (-1)^n + b$$. According to our previous analysis (see Theorems 3.1 and 3.2), these equations admit symmetries of order one and two in the $$m$$ and $$n$$ direction, respectively, and corresponding canonical conservation laws can be found in the Appendix. Moreover, solutions of these two equations are related according to

Proposition 5.1.
The system of equations
$$u_{n,m+1}-u_{n+1,m} =v_{n,m+1} ,\quad u_{n+1,m}-u_{n,m-1} = \frac{\alpha_n}{v_{n,m}}$$
(48)
defines a Bäcklund transformation between a solution $$v_{n,m}$$ of equation (46) and a solution $$u_{n,m}$$ of equation (47).
We are going to consider the $$(k,-1)$$ periodic reductions of equations (46) and (47). The choice of the steps in this reduction is motivated from the dependence of the equations on a periodic function of $$n$$. Such reductions amount to consider the equation along with the relation $$u_{n,m+1} = u_{n+k,m}$$ and its consequences, $$u_{n+i,m+j} = u_{n+i + j k,m}$$. These relations clearly allow us to remove any dependence on $$m$$ from the equation and derive a non-autonomous map with $$n$$ as the independent variable. Specifically, equation (46) becomes
$$v_{n}+ \frac{\alpha_n}{v_{n+k}} - \frac{\alpha_{n+1}}{v_{n+1}} -v_{n+k+1} =0 , \quad \alpha_{n+2} = \alpha_{n},$$
(49)
whereas equation (47) reduces to
$$(u_{n+k+1} - u_{n}) (u_{n+k}- u_{n+1}) = \alpha_{n},\quad \alpha_{n+2} = \alpha_{n}.$$
(50)

In both equations, we have suppressed the second index of functions $$v$$ and $$u$$ since it does not vary.

Moreover, we can replace shift operator $$\cal{T}$$ with $${\cal{S}}^k$$ in any conserved form of the original equations and potentially derive first integrals of the resulting maps. If $$({\cal{S}}-1)(p_n) = ({\cal{T}}-1)(q_n)$$ is a conserved form of the equation, then the corresponding candidate for a first integral will be $$p_n -q_{n}-\ldots-q_{n+k-1}$$. In the same way, we may derive a Lax pair for the reduced equations. More precisely, if system $$\Psi_{n+1,m} = L_{n,m} \Psi_{n,m}$$, $$\Psi_{n,m+1} = M_{n,m} \Psi_{n,m}$$ is the Lax pair of the original equation and $$L_{n,m+1} M_{n,m} = M_{n+1,m} L_{n,m}$$ is its compatibility condition, then in view of the $$(k,-1)$$ periodic reduction the latter relation becomes $$L_{n+k} M_{n} = M_{n+1} L_{n}$$, where we have omitted the second index. We may now define matrices $${\cal{M}}_n = L_{n+k-1} L_{n+k-2} \cdots L_{n} M_n^{-1}$$ and $${\cal{L}}_n = L_{n+k}$$, and then write the reduced compatibility condition as $${\cal{M}}_{n+1} = {\cal{L}}_n {\cal{M}}_n {\cal{L}}_n^{-1}$$.

It should be noted though that we can construct first integrals and Lax pairs in the way we described above only when $$k$$ is a multiple of two. This is because the equations are non-autonomous and function $$\alpha_{n}$$ is periodic with period $$2$$. As a consequence we separate the two cases below and first we discuss the even reductions $$(k=2 p)$$ for both equations and present their connections, and then we analyse the odd order reductions $$(k=2 p +1)$$.

Remark 5.2

The maps (49) may be considered a non-autonomous extension of the ones studied recently in [20]. Moreover, as a generalization of Proposition 4.2 in [20], one can easily prove

Proposition 5.3
Suppose that $$v_n = \tau_{n} \tau_{n+k+1}/(\tau_{n+1} \tau_{n+k})$$ is a solution of equation (49). Then, for any integer value of $$k$$, function $$\tau_{n}$$ satisfies the bilinear equation
$$b\, \tau_{n+k} \tau_{n+k+1} = \alpha_n \tau_{n+1} \tau_{n+2 k} + \tau_n \tau_{n+2 k +1},\quad \alpha_{n+2} = \alpha_{n} ,\,\, \alpha_{n+1} \ne \alpha_{n},\,\, b\in {\mathbb{R}}.$$
If $$k$$ is odd, then $$\tau_n$$ also satisfies
$$b^\prime_n \tau_{n+1} \tau_{n+k+1} = \alpha_{n+1} \tau_{n+2} \tau_{n+k} - \tau_n \tau_{n+k+2},\quad \alpha_{n+2} = \alpha_{n} ,\,\, \alpha_{n+1} \ne \alpha_{n},\,\, b^\prime_{n+k} = b^\prime_n.$$

### 5.1 The even order reductions

#### 5.1.1 Hirota’s KdV equation

If $$k =2 p$$, $$p\in {\mathbb{Z}}_{>0}$$, equation (49) becomes
$$v_{n}+ \frac{\alpha_n}{v_{n+2 p}} - \frac{\alpha_{n+1}}{v_{n+1}} - v_{n+2 p+1} =0 ,$$
(51)
and admits two first integrals and a two-integral which follow from our considerations and can be found in the Appendix, see relations (A.3A.5). Moreover one additional integral follows from the equation itself since it is a total difference yielding the first integral
$$J_n = v_{n} + \cdots + v_{n+2 p} - \sum_{i=1}^{2 p -1} \frac{\alpha_{n+i}}{v_{n+i}}.$$
(52)
For these maps we can construct matrices $${\cal{M}}_n$$ and $${\cal{L}}_n$$ using as building blocks
$$L_n= \left( \begin{array}{cc} -\,\frac{\alpha_{n}}{v_{n}} & \lambda \\ \lambda & v_{n} \end{array} \right)\quad {\mbox{and}} \quad M_{n} = \left( \begin{array}{cc} -\,\frac{\alpha_{n}}{v_{n}} - v_{n+2 p} & \lambda \\ \lambda &0 \end{array} \right)\!.$$

The trace of matrix $${\cal{M}}_n$$ is a polynomial of the form $$\sum_{i=0}^{p} F^{(i)}_n \lambda^{2 i-2}$$, and the coefficients of $$\lambda$$ are first integrals of the map. For the cases, we have checked we found $$F^{(p)}_n = J_n$$ and $$F^{(0)}_n = I_n$$, where $$J_n$$ and $$I_n$$ are given in (52) and (A.3), respectively.

Example 4
For $$p=1$$ (or equivalently $$k=2$$), the map is
$$v_{n}+ \frac{\alpha_n}{v_{n+2}} - \frac{\alpha_{n+1}}{v_{n+1}} - v_{n+3} =0,$$
(53)
and it admits two functionally independent first integrals,
$$J_n = v_n + v_{n+1} + v_{n+2} - \frac{\alpha_{n+1}}{v_{n+1}},\quad I_n =(\alpha_{n} + v_n v_{n+2}) v_{n+1}.$$
(54)
Example 5
For $$p=2$$ (or equivalently $$k=4$$), the map is
$$v_{n}+ \frac{\alpha_n}{v_{n+4}} - \frac{\alpha_{n+1}}{v_{n+1}} - v_{n+5} =0,$$
and it admits two first integrals,
$$J_n = v_n + v_{n+1} + v_{n+2} + v_{n+3}+v_{n+4} - \frac{\alpha_{n+1}}{v_{n+1}}- \frac{\alpha_{n}}{v_{n+2}}- \frac{\alpha_{n+1}}{v_{n+3}},\quad I_n =(\alpha_{n} + v_n v_{n+4}) v_{n+1} v_{n+2} v_{n+3},$$
and the two-integral $$T_n$$ (A.5), which in this case has the following form,
$$T_n = \frac{F_n F_{n+2}}{v_{n+2}}, \quad {\mbox{where}}\quad F_n = \alpha_{n} \left(\frac{\alpha_{n+1}}{v_{n+1}}-v_n\right) + v_{n+2} (v_n v_{n+1}-\alpha_{n}).$$
Two more integrals can be derived from our considerations, $$K^\prime_n$$ from the trace of the monodromy matrix, and $$K_n$$ from the general formula (A.4) in the Appendix. However, they are not functionally independent since
$$K^\prime_n = \frac{T_{n+1} +T_{n} -2 I_n - (\alpha_{n}^2+\alpha_{n+1}^2) J_n}{\alpha_{n}+\alpha_{n+1}} \quad {\mbox{and}} \quad K_n= \left(\frac{\alpha_{n} \alpha_{n+1}}{I_n}\right)^2 - \frac{K^\prime_n}{I_n} - \frac{1}{\alpha_{n+1}}-\frac{1}{\alpha_{n}}.$$

#### 5.1.2 The potential KdV equation

Considering equation (50) with $$k =2 p$$, we introduce a new variable $$w_{n}= u_{n+1}- u_{n}$$ to derive
$$\left(w_{n}+w_{n+1}+\cdots+w_{n+2 p} \right) \left( w_{n+1}+ w_{n+2}+\cdots+w_{n+2 p-1}\right)= \alpha_{n}.$$
(55)

First integrals follow from the reduction of the conserved forms of equation (47) and can be found in the Appendix, see relations (A.9A.11).

Example 6
The reduced difference equation for $$p=1$$$$(k=2)$$ is
$$(w_{n}+w_{n+1}+w_{n+2}) w_{n+1} = \alpha_{n},$$
(56)
which may be viewed as a degenerated form of (the most general) discrete Painlevé I [27], and admits the first-integral
$$I_n = w_n w_{n+1} (w_n+w_{n+1}) - \alpha_{n+1} w_{n+1} - \alpha_{n} w_n.$$
(57)

#### 5.1.3 Bäcklund transformations

One can derive connections among the maps constructed previously by applying the same periodic reduction to Proposition 5.1. It is not difficult to see that transformation (48) is compatible only with the even order reductions because of the periodicity of function $$\alpha_{n}$$. So we state the following

Proposition 5.4
The system of equations
$$v_{n+2 p} = w_{n+1} + \cdots + w_{n+2 p -1},\quad \frac{\alpha_{n}}{v_{n}} = w_{n-2p} + \cdots + w_n$$
(58)
defines a transformation between a solution $$v_n$$ of (51) and a solution $$w_n$$ of (55).
Proof.

If we shift the second equation $$2 p$$ times and multiply the resulting relation with the first equation in (58), then we find that $$w_n$$ satisfies (55). For the converse, let us denote with $$f$$ and $$g$$ the first and the second equation in (58), respectively. Then the combination $${\cal{S}}^{2 p}(g) - {\cal{S}}(f)$$ yields $$w_{n+1}+w_n = v_{n+2 p+1} - \alpha_{n}/v_{n+2 p}$$, whereas $${\cal{S}}(g) - {\cal{S}}^{-2 p}(f)$$ leads to $$w_{n+1}+w_n = v_{n} - \alpha_{n+1}/v_{n+1}$$. From the last two relations it follows that $$v_n$$ must obey (51). □

### 5.2 The odd order reductions

#### 5.2.1 Hirota’s KdV equation

When $$k =2 p+1$$ we introduce a new variable $$x_{n} = v_{n} v_{n+1}$$ in view of which map (49) becomes
$$\frac{x_n - \alpha_{n+1}}{x_{n+1} x_{n+3} \cdots x_{n+2 p -1}} - \, \frac{x_{n+2 p+1}-\alpha_n }{x_{n+2} x_{n+4} \cdots x_{n+2 p}} = 0.$$
(59)
This map admits the first integral
$$I_n = \prod_{i=0}^{p} x_{n+ 2 i}\,+ \, \alpha_n \,\prod_{i=0}^{p-1} x_{n+2 i+1},$$
(60)
which can be derived either from (59) or from the reduction of the first conservation law (A.2). We may consider $$I_n = 1$$ as our main equation which after the change $$x_n \mapsto -\alpha_{n}^{(p+1)/(2 p+1)} \alpha_{n+1}^{p/(2 p+1)} x_n$$ can be written as
$$\prod_{i=0}^{p} x_{n+ 2 i}\,- \, \prod_{i=0}^{p-1} x_{n+2 i+1} = b_n,$$
(61)
where $$b_n = (-1)^p \alpha_{n}^{-(p+1)^2/(2 p+1)} \alpha_{n+1}^{-p (p+1)/(2 p+1)}$$ is periodic function with period 2. If we introduce even and odd variables $$e_n$$ and $$o_n$$ via the relations $$e_{n+i}=x_{2 n+2 i}$$ and $$o_{n+i} = x_{2 n+2 i +1}$$, respectively, and denote $$b_{2 n+2i} = \alpha$$, $$b_{2 n+2 i +1} = \beta$$, then these variables will satisfy equations $$G_p(e_n,\alpha,\beta) =0$$ and $$G_p(o_n,\beta,\alpha)=0$$, respectively, where
$$G_p(f_n,a,b) = \prod_{j=0}^{p} \left(\prod_{i=-p}^{0} f_{n+i+j} -a \right)\,-\, \prod_{j=0}^{p-1} \left(\prod_{i=1-p}^{0} f_{n+i+j} + b \right)\!.$$
(62)
Example 7

For $$p=1$$, the map (61) becomes $$x_{n} x_{n+2} - x_{n+1}= b_{n}$$ which is the second order periodic non-autonomous Lyness’ difference equation, an integrable map [21]. The corresponding odd-even variables decoupled system is the QRT map $$(f_{n-1} f_{n}-a) (f_{n} f_{n+1} -a) = f_{n} + b$$.

Example 8
When $$p=2$$ the equation is
$$x_n x_{n+2} x_{n+4} - x_{n+1} x_{n+3} = b_n ,$$
(63)
and the corresponding decoupled system of odd-even variables is related to solutions of the fourth order equation
$$(\,f_{n-2} f_{n-1} f_{n}-a) (\,f_{n-1} f_{n} f_{n+1} -a) (\,f_{n} f_{n+1} f_{n+2} -a)= (\,f_{n-1} f_{n}+b) (\,f_{n} f_{n+1} +b).$$
(64)

#### 5.2.2 The potential KdV equation

When $$k=2 p +1$$ the left hand side of (50) can be written as $$({\cal{S}}^{2 p+1} +1) \left(u_{n} u_{n+1}\right) -({\cal{S}}+1) \left(u_{n} u_{n+2 p + 1}\right)$$. This allows us to integrate it once and then, upon the difference substitution $$x_{n} = (-1)^n u_{n} u_{n+1}$$, write the resulting equation as
$$x_{n} + \cdots + x_{n+2 p} - (-1)^p x_{n+2 p} \prod_{i=0}^{p-1} \frac{x_{n+2 i}}{x_{n+2 i+1}}\, = \frac{(-1)^n}{2} ({\cal{S}}-1) (n\,\alpha_n) +\gamma,$$
(65)
where $$\gamma$$ is the constant of integration. Since $$\alpha_{n}$$ is a periodic function with period 2, we can choose without loss of generality $$\alpha_{n} = -\alpha (-1)^n +2 \beta$$, change $$\gamma \rightarrow \gamma -\alpha/2$$ and then write equation (65) as
$$x_{n} + \cdots + x_{n+2 p} - (-1)^p x_{n+2 p} \prod_{i=0}^{p-1} \frac{x_{n+2 i}}{x_{n+2 i+1}}\, = \, z_n\,,\quad {\mbox{ where }} \quad z_n = \alpha \,n + \beta (-1)^n + \gamma,$$
(66)
i.e. a family of discrete Painlevé type equations.
Example 9
The first member $$(p=1)$$ of the family is $$(x_n +x_{n+1})(x_{n+1}+x_{n+2}) \,=\,z_n\, x_{n+1}$$ which, upon the substitution $$x_{n+1}+x_n= y_{n}$$ becomes $$x_n = y_n- y_n y_{n+1}/z_n$$ and then leads to
$$\frac{y_{n+2}}{z_{n+1}} + \frac{y_n}{z_n} = 1.$$

## 6. Concluding remarks

We presented a method for the deautonomization of partial difference and differential-difference equations using the existence of infinite hierarchies of symmetries in both directions as integrability detector. In this way, we derived integrable non-autonomous difference equations along with compatible differential-difference equations (symmetries) and conservation laws. Our results include non-autonomous versions of various known quad equations and their second and third order non-autonomous symmetries which, as far as we are aware, are new and deviate from previous works [22, 28, 29] in which the first symmetries are of order one and related to Yamilov’s discrete non-autonomous Krichever–Novikov equation [30]. The most interesting phenomenon is that most of the deautonomized equations and their symmetries are related via Miura transformations to the two-quad equation (8) and its symmetries (9). Our results along with the ones in [16, 17, 31] justify the importance of equation (9) as it was already pointed in [17].

Reductions of the derived differential-difference equations led to integrable non-autonomous ordinary difference equations and in certain cases to alternating QRT maps, whereas the reductions of the corresponding conservation laws yielded integrals for the resulting equations. And the periodic reductions of equations (46) and (47) led to families of non-autonomous maps and discrete Painlevé type equations.

It would be interesting to apply this method for the derivation of non-autonomous differential-difference equations of order higher than two (with one such example provided by (36)), but also to extend it to systems of difference equations and their symmetries.

## Acknowledgements

The author would like to thank Frank Nijhoff and Andy Hone for useful discussions and suggestions.

### Appendix

In this section, we have collected three canonical conservation laws and their reductions for equations (46) and (47).

• i) Canonical conservation laws for the non-autonomous Hirota KdV equation (46).
\begin{eqnarray} \left( \rho,\sigma\right) &=& \left( \log v_{n,m},\,\, \log \frac{v_{n,m} v_{n+1,m}}{\omega_{n+1,m}}\right)\quad {\mbox{with }}\,\, ({\cal{S}}^2-1)(\rho) = ({\cal{T}}-1)(\sigma), \\ \end{eqnarray}
(A.1)

\begin{eqnarray} \left( \varrho^{(0)},\varphi^{(0)} \right) &=& \left( \log \frac{v_{n,m}}{F_{(+)}}, \log v_{n,m}\right)\!,\quad \left( \varrho^{(1)},\varphi^{(1)} \right) = \left(\frac{2 v_{n,m} v_{n,m-1}}{F_{(+)} F_{(-)}}, \frac{v_{n,m} v_{n,m-1}-\alpha_{n}}{\alpha_{n} F_{(-)}} \right)\!, \end{eqnarray}
(A.2)
where $$F_{(\pm)}= v_{n,m\pm 1} v_{n,m} + \alpha_{n}$$ and $$({\cal{S}}-1)(\varrho^{(i)}) = ({\cal{T}}-1)(\varphi^{(i)})$$, $$i= 0,1$$.
Considering even order periodic reductions $$(2 p,-1)$$, the reduction of the first conservation law in (A.2) yields the first integral
$$I_n = (\alpha_{n} +v_n v_{n+2 p}) \prod_{i=1}^{2 p-1} v_{n+i},$$
(A.3)
and the second pair in (A.2) becomes
$$K_n = \frac{2 v_n v_{n-2 p}}{(v_n v_{n + 2 p } + \alpha_{n})(v_n v_{n -2 p } + \alpha_{n})} - \sum_{i=0}^{2 p -1} \frac{1}{\alpha_{n+i}}{\cal{S}}^i \left( \frac{v_n v_{n-2 p} - \alpha_{n}}{v_n v_{n-2 p} + \alpha_{n}} \right)\!,$$
(A.4)
where one has to use successively the map to eliminate the negative shifts of $$v_{n}$$ and derive the corresponding integral. Moreover, conservation law (A.1) leads to the two-integral $$(T_{n+2}=T_n)$$
$$T_n =v_{n} \prod_{i=0}^{p -1} {\cal{S}}^{2 i} \left(\frac{v_{n+2} v_{n+1} (v_{n+1} v_{n}- \alpha_n)- \alpha_n (v_{n+1} v_{n}- \alpha_{n+1})}{v_{n} v_{n+1}} \right)\!.$$
(A.5)
• ii) Canonical conservation laws for the non-autonomous H1 equation (47).
\begin{eqnarray} \left(\tilde{\rho},\tilde{\sigma} \right) &=& \left(\log (u_{n,m+1}-u_{n-1,m}),\,\, \log ( h_{n,m})\right),\quad {\mbox{with }}\,\, ({\cal{S}}^2-1)(\tilde{\rho}) = ({\cal{T}}-1)(\tilde{\sigma}), \\ \end{eqnarray}
(A.6)

\begin{eqnarray} \tilde{\varrho}^{(0)} &=& \log G_{n,m},\quad \tilde{\varphi}^{(0)} = \log (u_{n+1,m}-u_{n,m-1}), \\ \end{eqnarray}
(A.7)

\begin{eqnarray} \tilde{\varrho}^{(1)} &=& \frac{-2}{G_{n,m} G_{n,m-1}},\quad \tilde{\varphi}^{(1)} = \frac{1}{G_{n,m-1}^2} \left( \frac{u_{n+1,m-1}-u_{n,m-2}}{u_{n,m-1}-u_{n+1,m-2}} + \frac{u_{n,m}-u_{n+1,m-1}}{u_{n,m-1}-u_{n+1,m}}\right)\!, \end{eqnarray}
(A.8)
where $$G_{n,m} := u_{n,m+1}- u_{n,m-1}$$ and $$({\cal{S}}-1)(\tilde{\varrho}^{(i)}) = ({\cal{T}}-1)(\tilde{\varphi}^{(i)})$$, $$i= 0,1$$.
The even order periodic reductions $$(2 p,-1)$$ of the conservation laws (A.6) and (A.7) yield the two-integral
$$T_n = \left(\prod_{i=0}^{p-1} \left((w_{n+2 i +2} +w_{n+2 i +1})(w_{n+2 i +1}+w_{n+2 i}) + \alpha_{n+1}-\alpha_{n} \right)\right) \sum_{j=1}^{2 p-1}w_{n+j}$$
(A.9)
and the first integral
$$I_n = \prod_{i=0}^{2p-1} {\cal{S}}^i \left(\sum_{j=0}^{2 p} w_{n+j}\right){\Big{/}}\sum_{\ell=0}^{4 p-1}w_{n+\ell}$$
(A.10)
for the reduced map (55). Finally the pair (A.8) generates the first integral
$$K_n = \frac{2}{\left(\sum_{i=0}^{4 p-1} w_{n+i}\right)\left(\sum_{i=2p}^{6 p-1} w_{n+i}\right) } + \sum_{i=0}^{2p-1} {\cal{S}}^i \left(\left(\sum_{j=0}^{4p-1} w_{n+j}\right)^{-2} \left(\frac{\sum_{s=0}^{2p} w_{n+s}}{\sum_{r=1}^{2p-1} w_{n+r}}- \frac{\sum_{s=2p+1}^{4p-1} w_{n+s}}{\sum_{r=2p}^{4p} w_{n+r}}\right)\right).$$
(A.11)

In the first integrals (A.9)–(A.11) we must use equation (55) and its shifts to eliminate the values $$w_{n+\ell}$$ with $$\ell> 2p$$.

## Footnotes

1 In principle, this approach works also with Q4 but we could not verify it due to computational limitations.

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