Abstract. The main physical results of this paper are new exact analytical solutions of the heavenly equation, of importance in the general theory of relativity. These solutions are not invariant under any subgroup of the symmetry group of the equation. The main mathematical result is a new method of obtaining non-invariant solutions of partial differential equations with infinite-dimensional symmetry groups. The method involves the compatibility of the given equations with a differential constraint, which is automorphic under a specific symmetry subgroup, the latter acting transitively on the submanifold of the common solutions. By studying the integrability of the resulting conditions, one can provide an explicit foliation of the entire solution manifold of the considered equations.

PACS numbers: 02.20.Tw, 02.30.Jr, 04.20.Jb, 03.65.Fd

Print publication: Issue 43 (2 November 2001)
Received 19 April 2001
Published 19 October 2001