### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 075, 21 pages      arXiv:1306.3195      https://doi.org/10.3842/SIGMA.2013.075

### Partner Symmetries, Group Foliation and ASD Ricci-Flat Metrics without Killing Vectors

Mikhail B. Sheftel a and Andrei A. Malykh b
a) Department of Physics, Boğaziçi University 34342 Bebek, Istanbul, Turkey
b) Department of Numerical Modelling, Russian State Hydrometeorlogical University, 98 Malookhtinsky Ave., 195196 St. Petersburg, Russia

Received June 14, 2013, in final form November 19, 2013; Published online November 27, 2013

Abstract
We demonstrate how a combination of our recently developed methods of partner symmetries, symmetry reduction in group parameters and a new version of the group foliation method can produce noninvariant solutions of complex Monge-Ampère equation (CMA) and provide a lift from invariant solutions of CMA satisfying Boyer-Finley equation to non-invariant ones. Applying these methods, we obtain a new noninvariant solution of CMA and the corresponding Ricci-flat anti-self-dual Einstein-Kähler metric with Euclidean signature without Killing vectors, together with Riemannian curvature two-forms. There are no singularities of the metric and curvature in a bounded domain if we avoid very special choices of arbitrary functions of a single variable in our solution. This metric does not describe gravitational instantons because the curvature is not concentrated in a bounded domain.

Key words: Monge-Ampère equation; Boyer-Finley equation; partner symmetries; symmetry reduction; non-invariant solutions; group foliation; anti-self-dual gravity; Ricci-flat metric.

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