# On a classification algorithm of the integrable two-dimensional lattices via Lie-Rinehart algebras

## TL;DR

This paper develops an algebraic classification method for integrable nonlinear lattices with one discrete and two continuous variables, using Lie-Rinehart algebras to identify integrability conditions.

## Contribution

It introduces a new classification algorithm based on characteristic Lie-Rinehart algebras for integrable lattices, providing new results in the field.

## Key findings

- Classification algorithm based on characteristic algebra properties
- Identification of finite-dimensional Lie-Rinehart algebras for integrability
- Some new classification results for nonlinear lattices

## Abstract

In the article the problem of the integrable classification of nonlinear lattices depending on one discrete and two continuous variables is studied. By integrability we mean the presence of reductions of a chain to a system of hyperbolic equations of arbitrarily high order integrable in the Darboux sense. Darboux integrablity admits a remarkable algebraic interpretation: the Lie-Rinehart algebras related to both characteristic directions corresponding to the reduced system of the hyperbolic equations have to be of finite dimension. A classification algorithm based on the properties of the characteristic algebra is discussed. Some classification results are presented.