### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 066, 13 pages      arXiv:1311.2240      https://doi.org/10.3842/SIGMA.2014.066

### Non-Point Invertible Transformations and Integrability of Partial Difference Equations

Sergey Ya. Startsev
Ufa Institute of Mathematics, Russian Academy of Sciences, 112 Chernyshevsky Str., Ufa, 450077, Russia

Received November 10, 2013, in final form June 11, 2014; Published online June 17, 2014

Abstract
This article is devoted to the partial difference quad-graph equations that can be represented in the form $\varphi (u(i+1,j),u(i+1,j+1))=\psi (u(i,j),u(i,j+1))$, where the map $(w,z) \rightarrow (\varphi(w,z),\psi(w,z))$ is injective. The transformation $v(i,j)=\varphi (u(i,j),u(i,j+1))$ relates any of such equations to a quad-graph equation. It is proved that this transformation maps Darboux integrable equations of the above form into Darboux integrable equations again and decreases the orders of the transformed integrals by one in the $j$-direction. As an application of this fact, the Darboux integrable equations possessing integrals of the second order in the $j$-direction are described under an additional assumption. The transformation also maps symmetries of the original equations into symmetries of the transformed equations (i.e. preserves the integrability in the sense of the symmetry approach) and acts as a difference substitution for symmetries of a special form. The latter fact allows us to derive necessary conditions of Darboux integrability for the equations defined in the first sentence of the abstract.

Key words: quad-graph equation; non-point transformation; Darboux integrability; higher symmetry; difference substitution; discrete Liouville equation.

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