On some exact solutions of heavenly equations in four dimensions

TL;DR

This paper presents new exact, non-invariant solutions to various heavenly equations in four dimensions, leading to metrics without symmetries, and investigates their invariance properties.

Contribution

It introduces new classes of functionally-invariant solutions to multiple heavenly equations and provides criteria for their non-invariance, expanding understanding of four-dimensional metrics.

Findings

Found new classes of solutions for complex Monge-Ampère and heavenly equations.

Identified solutions that correspond to metrics without Killing vectors.

Developed criteria to determine the non-invariance of these solutions.

Abstract

Some new classes of exact solutions (so-called functionally-invariant solutions) of the elliptic and hyperbolic complex Monge-Amp$\grave{e}$re equations and of the second heavenly equation, mixed heavenly equation, asymmetric heavenly equation, evolution form of second heavenly equation, general heavenly equation, real general heavenly equation and one of the real sections of general heavenly equation, are found. Besides non-invariance of these found classes of solutions has been investigated. These classes of solutions determine the new classes of metrics without Killing vectors. A criterion of non-invariance of the solutions belonging to found classes, has been also formulated.