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

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.