Reductions of the universal hierarchy and rdDym equations and their symmetry properties

TL;DR

This paper investigates specific reductions of the universal hierarchy and rdDym equations, analyzing their symmetry properties and describing the associated Lie algebras of nonlocal symmetries within infinite-dimensional coverings.

Contribution

It provides a detailed analysis of the symmetry structures of these reduced equations and characterizes their nonlocal Lie algebras, advancing understanding of their integrability properties.

Findings

Identification of nonlocal symmetry Lie algebras

Description of symmetry properties in infinite-dimensional coverings

Connection between reductions and universal hierarchy

Abstract

We consider the equations $u_{yy} = u_y u_{xx} - (u_x + u)u_{xy} + u_x u_y$ and $u_{yy} = (u_x + x)u_{xy} - (u_{xx} + 2)u_y$ that arise as reductions of the universal hierarchy and rdDym equations, respectively, and describe the Lie algebras of nonlocal symmetries in infinite-dimensional coverings naturally associated to these equations.