On some exact solutions of heavenly equations in four dimensions

Abstract

Some new classes of exact solutions (so-called functionally invariant solutions) of the elliptic and hyperbolic complex Monge-Ampère equations, the second heavenly equation, mixed heavenly equation, asymmetric heavenly equation and evolution form of second 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 heavenly metrics without Killing vectors. A criterion of non-invariance of the solutions belonging to found classes, has been also formulated.