# Classification of a Subclass of Two-Dimensional Lattices via Characteristic Lie Rings

## TL;DR

This paper introduces a new classification algorithm applied to identify integrable cases of a specific subclass of two-dimensional lattices, confirming that only one known lattice satisfies the integrability criteria.

## Contribution

The paper develops a novel classification method based on characteristic Lie rings and applies it to a subclass of lattices, successfully identifying the unique integrable lattice within this class.

## Key findings

- Only one lattice passes the Darboux integrability test.
- The identified lattice is the Ferapontov-Shabat-Yamilov equation.
- The reduced lattice also passes the symmetry integrability test.

## Abstract

The main goal of the article is testing a new classification algorithm. To this end we apply it to a relevant problem of describing the integrable cases of a subclass of two-dimensional lattices. By imposing the cut-off conditions $u_{-1}=c_0$ and $u_{N+1}=c_1$ we reduce the lattice $u_{n,xy}=\alpha(u_{n+1},u_n,u_{n-1})u_{n,x}u_{n,y}$ to a finite system of hyperbolic type PDE. Assuming that for each natural $N$ the obtained system is integrable in the sense of Darboux we look for $\alpha$. To detect the Darboux integrability of the hyperbolic type system we use an algebraic criterion of Darboux integrability which claims that the characteristic Lie rings of such a system must be of finite dimension. We prove that up to the point transformations only one lattice in the studied class passes the test. The lattice coincides with the earlier found Ferapontov-Shabat-Yamilov equation. The one-dimensional reduction $x=y$ of this lattice passes also the symmetry integrability test.