Using the formalism of complex ℋ-spaces, we show that all real, Euclidean self-dual spaces that admit (at least) one Killing vector may be gauged so that only two distinct types of Killing vectors appear; in Kähler coordinates these are the generators of a translational or a rotational symmetry. We give explicit forms both for the Killing vectors and for the constraint on the Kähler potential function Ω which allows for such a Killing vector. In the translational case we show how all such spaces are determined by the general solution of the three-dimensional, flat Laplace’s equation and how these are related to the multi-Taub–NUT metrics of Gibbons and Hawking. In the rotational case we simplify the equation determining Ω, but this is not sufficient to obtain the general solution.

Following Plebanski and Robinson, complex V.s which admit a congruence of totally null surfaces are shown to have coordinates which, in pairs, have a spinor structure which generates the usual spinor structure of the 2-forms over the space. This structure allows Einstein's vacuum equations to fracture into three triples and a singlet, which allow for easy reduction of the entire set to one nonlinear partial differential equation needed for consistency. An inhomogeneous GL(2, C) group of coordinate transformations, constrained to leave the tetrad form invariant, is constructed and used to simplify the equations and clarify the geometrical meaning of the parameters introduced during the integration process.

We show that all ,/Y' 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 cW' 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.

New developments in a continuing investigation of complex V,s with purely self-dual conformal curvature are presented: (I) conformal and projective extensions of spaces with CABer, = ° are discussed; (2) Killing vectors for general heavenly metrics are determined; (3) the solutions, in heavenly spaces, for (massless) D(O,s) spin or fields (in particular, the Maxwell field) are found; then (4) new examples of heavenly metrics of types Gx [-) and DX H are provided; lastly, contraction of the DX D solutions of Pleban'ski and Demianski to type DX H is performed, giving a complex prototype of the Kerr-Newman solution, and all solutions of the type NX H are given, which contain two arbitrary functions of two variables.