Invariant solutions to complex Monge-Ampere equation and gravitational instantons

Generalized gravitational instanton solutions to the self-dual Einstein fie ld equations with one Killing vector are studied by using symmetry-group an alysis. The starting point is an earlier classification of the subgroups of the symmetry group of the complex Monge-Ampere equation for the Kahler pot ential of the metric. Two of the four nonequivalent one-dimensional subgrou ps are used to obtain reductions to equations in three real dimensions. One of the reduced equations is linearized via a contact transformation. A Bac kground transformation obtained for the other one relates the "heavenly equ ation" to another system of nonlinear equations.