ar X iv : g r - qc / 0 10 50 88 v 1 2 4 M ay 2 00 1 A family of heavenly metrics

The antecedent of Einstein field equations with Euclidean signature is the complex elliptic Monge-Ampère equation, CMA2. For elliptic CMA2 solutions determine hyper-Kähler, self-dual and therefore Ricci-flat metrics with Euclidean signature. We shall consider a symmetry reduction of CMA2 to solutions that admit only a single Killing vector. Systematic studies of vacuum metrics with one Killing vector showed that they fall into two classes depending on whether the Killing vector is translational, or rotational. In the first case we have Gibbons-Hawking metrics [1] where the Einstein field equations reduce to Laplace’s equation, whereas in the latter it is the heavenly equation [2, 3] for which up to the present no non-trivial exact solutions were known. In both cases the 2-form derived from the derivative of the Killing vector is self-dual [3, 4, 5, 6]. We refer to [7] and [8] for the H-space context of these solutions. So far all solutions of the heavenly equation discussed in the literature consist of invariant solutions, i.e. these solutions are invariant with respect to one-parameter symmetry subgroups of the heavenly equation itself. This symmetry carries over into the metric in the form of extra Killing vectors. We shall present metrics derived from non-invariant solutions of the heavenly equation which will not possess any new symmetries. In section 2 we shall briefly recall the reduction of CMA2 to heavenly equation. We construct a new family of hyper-Kähler metrics with one rotational Killing vector generated by non-invariant solutions of the heavenly