Dispersionless Hirota system and hidden symmetries of heavenly equation

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 $f\mapsto Φ(f)$ 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 Plebański 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 $f\mapsto Φ(f)$ mentioned and obtain explicit formulas for the metric depending on $Φ$. We present some examples showing how the Weyl spinor changes along with $Φ$.