# Self-Dual Systems, their Symmetries and Reductions to the Bogoyavlensky Lattice

## TL;DR

This paper explores self-dual discrete integrable systems with ${\mathbb{Z}}_N$ grading, analyzing their continuous symmetries, reductions, and connections to the Bogoyavlensky lattice, advancing understanding of their structure and integrability.

## Contribution

It introduces a class of self-dual systems within ${\mathbb{Z}}_N$ graded discrete Lax pairs and investigates their symmetries and reductions related to the Bogoyavlensky lattice.

## Key findings

- Identification of continuous symmetries of self-dual systems
- Establishment of reductions leading to Bogoyavlensky equations
- Deeper understanding of the structure of ${\mathbb{Z}}_N$ graded integrable systems

## Abstract

We recently introduced a class of ${\mathbb{Z}}_N$ graded discrete Lax pairs and studied the associated discrete integrable systems (lattice equations). In particular, we introduced a subclass, which we called "self-dual". In this paper we discuss the continuous symmetries of these systems, their reductions and the relation of the latter to the Bogoyavlensky equation.