All ASD complex and real 4-dimensional Einstein spaces with Λ≠0 admitting a nonnull Killing vector

Abstract

Anti-self-dual (ASD) 4-dimensional complex Einstein spaces with nonzero cosmological constant [Formula: see text] equipped with a nonnull 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–Plebański equation (Toda field equation). Some alternative forms of the metric are discussed. All possible real slices (neutral, Euclidean and Lorentzian) of ASD complex Einstein spaces with [Formula: see text] admitting a nonnull Killing vector are found.