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 $f \to \phi(f)$ of the Hirota system essentially changes the properties of the corresponding metric.

In this paper we restate these observations in the context of II Plebański heavenly equation (IIPHE). Namely, we first extend to 5D the hierarchy for this equation found by Dunajski and Mason in 2000 for even dimensions. We then consider the corresponding 5D system with a special type of symmetry generalizing the tri-holomorphic symmetry of IIPHE. 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 IIPHE. Such a point of view allows to reinterpret the symmetry $f \to \phi(f)$ mentioned and obtain explicit formulas for the metric depending on $\phi$. We present some examples showing how the Weyl spinor changes along with $\phi$.
(joint work with Adam Szereszewski)