The classification of all ℋ spaces admitting a Killing vector

Abstract

We show that all ℋ spaces (self-dual solutions of the complex Einstein vacuum equations) that admit (at least) one Killing vector may be gauged in such a way as to be divided into only five types, characterized by the type of equation which determines their potential function. In four of these types we show that this knowledge is sufficient to reduce the requirement of being an ℋ space to a linear equation whose solutions are well known. The fifth case is reduced considerably and a large class of special solutions is given.