We consider four-dimensional, Riemannian, Ricci-flat metrics for which one or other of the self-dual or anti-self-dual Weyl tensors is type-D in the Petrov-Pirani-Penrose classification. Such metrics always admit a valence-2 Killing spinor, and therefore a Hermitian structure and at least one Killing vector. We rederive the results of Przanowski and collaborators, [14-17], 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, [22], can be applied to show that the metrics can also be given in terms of an axisymmetric solution of the flat three-dimensional Laplacian. Thus in particular the field equations linearise. As a corollary, we show that the same technique linearises the field equations for a four-dimensional Einstein metric with anti-self-dual Weyl tensor and two commuting symmetries. We reduce the Einstein equations with non-zero scalar curvature and one-sided type-D Weyl tensor, excluding the Kahler-Einstein case, to a modified but not integrable Toda equation. Some examples of the constructions are given.

Thanks to the Institut Mittag-Leffler in Djursholm, Sweden for hospitality as part of the programme "General Relativity, Geometry and Analysis" during September 2019, supported by the Swedish Research Council under grant no. 2016-06596.