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 ...