We consider four-dimensional, Riemannian metrics for which one or other of the self-dual or anti-self-dual Weyl tensors is type-D and which satisfy the Einstein-Maxwell equations with the corresponding duality Maxwell field aligned with the type-D Weyl spinor, in the sense of sharing the same Principal Null Directions (or PNDs). Such metrics always have a valence-2 Killing spinor, and therefore a Hermitian structure and at least one Killing vector. We rederive the results of Araneda ([5]), that these metrics can all be given in terms of a solution of the SU(∞)-Toda field equation, and show that, when there is a second Killing vector commuting with the first, the method of Ward ([25]) can be applied to show that the metrics can also be given in terms of a pair of axisymmetric solutions of the flat three-dimensional Laplacian. Thus in particular the field equations linearise. Some examples of the constructions are given.

Thanks to the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme "Twistors in Geometry and Physics" (TWT) where work on this paper was undertaken. This work was supported by EPSRC grant no EP/R014604/34.