Higher symmetries of the lattices in 3D

TL;DR

This paper explores higher symmetries of 3D lattices, introduces a novel lattice not related by known transformations, and derives a new coupled DS-type system, expanding understanding of integrable lattice structures.

Contribution

It identifies a new 3D lattice with higher symmetries and derives a novel coupled DS-type system, not related to known lattices by Miura transformations.

Findings

Discovery of a new lattice with higher symmetries

Derivation of a new coupled DS-type system

Extension of duality between coupled systems and lattices

Abstract

It is known that there is a duality between the Davey--Stewartson type coupled systems and a class of integrable two--dimensional Toda type lattices. More precisely, the coupled systems are generalized symmetries for the lattices and the lattices can be interpreted as dressing chains for the systems. In our recent study we have found a novel lattice which apparently is not related to the known ones by Miura type transformation. In the article we described higher symmetries to this lattice and derived a new coupled system of the DS type.