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The main physical result of this paper are exact analytical solutions of theheavenly equation, of importance in the general theory of relativity. Thesesolutions are not invariant under any subgroup of the symmetry group of theequation. The main mathematical result is a new method of obtainingnoninvariant solutions of partial differential equations with infinitedimensional symmetry groups. The method involves the compatibility of the givenequations with a differential constraint, which is automorphic under a specificsymmetry subgroup, the latter acting transitively on the submanifold of thecommon solutions. By studying the integrability of the resulting conditions,one can provide an explicit foliation of the entire solution manifold of theconsidered equations.Comment: 32 pages, Latex, corrected typos, submitted to J. Phys. A: Math. Gen.  on April 13, 200

Group foliation and non-invariant solutions of the heavenly equation