Killing-Yano tensor and supersymmetry of the self-dual Plebanski-Demianski solution

We explore various aspects of the self-dual Pleba\'nski-Demia\'nski family in the Euclidean Einstein-Maxwell-$\Lambda$ system. The Killing-Yano tensor which was recently found by Yasui and one of the present authors allows us to prove that the self-dual Pleba\'nski-Demia\'nski metric can be brought into the self-dual Carter metric by an orientation-reversing coordinate transformation. We show that the self-dual Pleba\'nski-Demia\'nski solution admits two independent Killing spinors in the framework of $N=2$ minimal gauged supergravity, whereas the non-self-dual solution admits only a single Killing spinor. This can be demonstrated by casting the self-dual Pleba\'nski-Demia\'nski metric into two distinct Przanowski-Tod forms. As a by-product, a new example of the three-dimensional Einstein-Weyl space is presented. We also prove that the self-dual Pleba\'nski-Demia\'nski metric falls into two different Calderbank-Pedersen families, which are determined by a single function subjected to a linear equation on the two dimensional hyperbolic space. Furthermore, we consider the hyper-K\"ahler case for which the metric falls into the Gibbons-Hawking class. We find that the condition for the nonexistence of Dirac-Misner string enforces the solution with a nonvanishing acceleration parameter to the Eguchi-Hanson space.