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# All ASD complex and real 4-dimensional Einstein spaces with $$\Lambda \ne 0$$ admitting a nonnull Killing vector

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### Abstract

Anti-self-dual (ASD) 4-dimensional complex Einstein spaces with nonzero cosmological constant $$\Lambda$$ equipped with a nonnul Killing vector are considered. It is shown, that any conformally nonflat metric of such spaces can be always brought to a special form and the Einstein field equations can be reduced to the Boyer-Finley-Pleba\'nski equation (Toda field equation). Some alternative form of the metric are discussed. All possible real slices (neutral, Euclidean and Lorentzian) of ASD complex spaces admitting a nonnull Killing vector are found.

### Author and article information

###### Journal
2013-12-16
2015-02-02
###### Article
10.1142/S0219887816500110
1312.4284