Generalized symmetry classification of discrete equations of a class depending on twelve parameters

TL;DR

This paper classifies a broad class of discrete equations based on their symmetries, identifying integrable cases and their properties, including some that appear new but are related to known integrable equations.

Contribution

It provides a comprehensive symmetry classification of twelve-parameter polylinear discrete equations, revealing integrable examples and their structural properties.

Findings

Identified all equations linearizable via two-point first integrals.

Discovered new integrable examples with unique symmetry structures.

Enumerated equations with Darboux integrability and their relation to known equations.

Abstract

We carry out the generalized symmetry classification of polylinear autonomous discrete equations defined on the square, which belong to a twelve-parametric class. The direct result of this classification is a list of equations containing no new examples. However, as an indirect result of this work we find a number of integrable examples pretending to be new. One of them has a nonstandard symmetry structure, the others are analogues of the Liouville equation in the sense that those are Darboux integrable. We also enumerate all equations of the class, which are linearizable via a two-point first integral, and specify the nature of integrability of some known equations.