  The main physical result of this paper are 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
noninvariant 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.
