# On consistent systems of difference equations

## TL;DR

This paper develops a framework for analyzing overdetermined, consistent difference equations, constructing integrable hierarchies linked to known lattice systems, and exploring their symmetries and transformations.

## Contribution

It introduces a general approach for studying consistent difference systems and constructs two integrable hierarchies connected to well-known lattice equations.

## Key findings

- Constructed two hierarchies of consistent systems.
- Established relations to Bogoyavlensky and Sawada-Kotera lattices.
- Demonstrated integrability via higher order symmetries.

## Abstract

We consider overdetermined systems of difference equations for a single function $u$ which are consistent, and propose a general framework for their analysis. The integrability of such systems is defined as the existence of higher order symmetries in both lattice directions and various examples are presented. Two hierarchies of consistent systems are constructed, the first one using lattice paths and the second one as a deformation of the former. These hierarchies are integrable and their symmetries are related via Miura transformations to the Bogoyavlensky and the discrete Sawada-Kotera lattices, respectively.