Dispersionless Hirota system and hidden symmetries of heavenly equation

TL;DR

This paper explores the connection between dispersionless Hirota systems and heavenly equations, revealing hidden symmetries and their impact on self-dual vacuum Einstein metrics, with new 5D analogues and explicit metric formulas.

Contribution

It introduces 5D analogues of heavenly equations, analyzes a generalized symmetry, and provides explicit metric formulas depending on this symmetry, deepening understanding of the geometric structures.

Findings

Identification of 5D analogues of heavenly equations.

Explicit formulas for metrics depending on symmetry transformations.

Illustrations of Weyl spinor changes along with symmetry functions.

Abstract

In 2021 Konopelchenko, Schief and Szereszewski observed that solutions of 4D dispersionless Hirota system also solve the general heavenly equation describing self-dual vacuum Einstein metrics in neutral signature. They also noticed that the symmetry $f\mapsto \Phi(f)$ of the Hirota system essentially changes the properties of the corresponding metric. In this paper we restate these observations in the context of I and II Pleba\'nski heavenly equation (I,II PHE). Namely, we first find 5D analogues of these equations. We then consider a special type of symmetry generalizing the so-called tri-holomorphic symmetry of I or II PHE. The reduction with respect to this symmetry (which in a sense imitates the reduction of self-dual vacuum Einstein metrics with respect to a tri-holomorphic symmetry ending in special Einstein--Weyl structures) gives an analogue of the dispersionless Hirota system for I and II PHE. Such a point of view allows to reinterpret the symmetry $f\mapsto \Phi(f)$ mentioned and obtain explicit formulas for the metric depending on $\Phi$. We present some examples showing how the Weyl spinor changes along with $\Phi$.