# Tetrahedron maps and symmetries of three dimensional integrable discrete equations

## TL;DR

This paper explores the connection between tetrahedron maps and three-dimensional integrable discrete equations, introducing new examples and generalizing symmetry-based methods from Yang-Baxter maps.

## Contribution

It generalizes a symmetry-invariant method to relate tetrahedron maps with integrable lattice equations, producing new examples and insights.

## Key findings

- New tetrahedron maps discovered
- Established link between tetrahedron equations and lattice integrability
- Extended symmetry-based methods to 3D discrete systems

## Abstract

A relationship between the tetrahedron equation for maps and the consistency property of integrable discrete equations on $\mathbb{Z}^3$ is investigated. Our approach is a generalization of a method developed in the context of Yang-Baxter maps, based on the invariants of symmetry groups of the lattice equations. The method is demonstrated by a case-by-case analysis of the octahedron type lattice equations classified recently, leading to some new examples of tetrahedron maps and integrable coupled lattice equations.