On the algebra of nonlocal symmetries for the 4D Mart\'{\i}nez Alonso-Shabat equation

TL;DR

This paper analyzes the nonlocal symmetries of the 4D Martínez Alonso-Shabat equation, providing a comprehensive description of their Lie algebra structure using Lax pairs and differential coverings.

Contribution

It introduces a complete characterization of the Lie algebras of nonlocal symmetries for the 4D equation, extending previous results and embedding the hierarchy of symmetries into a larger algebra.

Findings

Describes two infinite-dimensional differential coverings.

Provides a complete Lie algebra structure of nonlocal symmetries.

Generalizes previous symmetry results to a larger algebraic framework.

Abstract

We consider the 4D Mart\'{\i}nez Alonso-Shabat equation $u_{ty} = u_z u_{xy} - u_y u_{xz}$ (also referred to as the universal hierarchy equation) and using its known Lax pair construct two infinite-dimensional differential coverings over $\mathcal{E}$. In these coverings, we give a complete description of the Lie algebras of nonlocal symmetries. In particular, our results generalize the ones obtained in [O.I.Morozov, A.Sergyeyev, The four-dimensional Mart\'{\i}nez Alonso-shabat equation: reductions and nonlocal symmetries. J. of Geom. and Phys. 85 (2014), 40--45 (arXiv:1401.7942v2)] and contain the constructed there infinite hierarchy of commuting symmetries as a subalgebra in a much bigger Lie algebra.