The symmetry structure of the anti-self-dual Einstein hierarchy

TL;DR

This paper explores the symmetry structure of the anti-self-dual Einstein equations, constructing a hierarchy of symmetries, a recursion operator, and a Lax pair, advancing understanding of this integrable system.

Contribution

It introduces a recursion operator and constructs the symmetry hierarchy and Lax pair for the anti-self-dual Einstein equations, revealing their algebraic structure.

Findings

Recursion operator for the hierarchy identified

Hierarchy of symmetries constructed

Lax pair explicitly formulated

Abstract

An important example of a multi-dimensional integrable system is the anti-self-dual Einstein equations. By studying the symmetries of these equations, a recursion operator is found and the associated hierarchy constructed. Owing to the properties of the recursion operator one may construct a hierarchy of symmetries and find the algebra generated by them. In addition, the Lax pair for this hierarchy is constructed.