Four-dimensional neutral signature self-dual gradient Ricci solitons

We describe the local structure of self-dual gradient Ricci solitons in neutral signature. If the Ricci soliton is non-isotropic then it is locally conformally flat and locally isometric to a warped product of the form $I\times_\varphi N(c)$, where $N(c)$ is a space of constant curvature. If the Ricci soliton is isotropic, then it is locally isometric to the cotangent bundle of an affine surface equipped with the Riemannian extension of the connection, and the Ricci soliton is described by the underlying affine structure. This provides examples of self-dual gradient Ricci solitons which are not locally conformally flat.