Twistor bundles, Einstein equations and real structures

We consider S-2 bundles P and P' of totally null planes of maximal dimension and opposite self-duality over a four-dimensional manifold equipped with a Weyl or Riemannian geometry. The fibre product PP' of P and P' is found to be appropriate for the encoding of both the self-dual and the Einstein-Weyl equations for the 4-metric. This encoding is realized in terms of the properties of certain well defined geometrical objects on PP'. The formulation is suitable for complex-valued metrics and unifies results for all three possible real signatures. In the purely Riemannian positive-definite case it implies the existence of a natural almost Hermitian structure on PP' whose integrability conditions correspond to the self-dual Einstein equations of the 4-metric. All Einstein equations for the 4-metric are also encoded in the properties of this almost Hermitian structure on PP'.