On the algebra of nonlocal symmetries for the 4D Mart\'ınez Alonso-Shabat equation

Abstract

We consider the 4D Mart\'ınez 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\'ınez 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.

arXiv comment14 pages

References in corpus (6)

• Hydrodynamic reductions and solutions of a universal hierarchy
• Energy-dependent potentials revisited: A universal hierarchy of hydrodynamic type
• The four-dimensional Martinez Alonso--Shabat equation: reductions and nonlocal symmetries
• The four-dimensional Martinez Alonso - Shabat equation: differential coverings and recursion operators
• Nonlocal symmetries, conservation laws, and recursion operators of the Veronese web equation
• Integrable systems in 4D associated with sixfolds in Gr(4,6)

Cited by in corpus (1)

• Extensions of the symmetry algebra and Lax representations for the two-dimensional Euler equation