Partner symmetries and non-invariant solutions of four-dimensional heavenly equations

TL;DR

This paper develops a method using partner symmetries to find explicit non-invariant solutions of four-dimensional heavenly equations, leading to new metrics without symmetries.

Contribution

It extends partner symmetry techniques to hyperbolic Monge-Ampère and second heavenly equations, producing non-invariant solutions via Legendre transformations.

Findings

Derived relations between partner symmetries for both equations

Transformed differential constraints into linear systems

Obtained explicit non-invariant heavenly metrics

Abstract

We extend our method of partner symmetries to the hyperbolic complex Monge-Amp\`ere equation and the second heavenly equation of Pleba\~nski. We show the existence of partner symmetries and derive the relations between them for both equations. For certain simple choices of partner symmetries the resulting differential constraints together with the original heavenly equations are transformed to systems of linear equations by an appropriate Legendre transformation. The solutions of these linear equations are generically non-invariant. As a consequence we obtain explicitly new classes of heavenly metrics without Killing vectors.