The Plebanski-Demianski (PD) solution has played a central role in the development of black-hole physics in general relativity since it describes the most general Petrov D metric in Einstein-Maxwell-Lambda system. Recently, Houri and Yasui have found a new king of nondegenerate rank-2 Killing-Yano (KY) tensor in the self-dual Euclidean PD metric. I will discuss in detail how this makes consistent with the theorem by Houri-Oota-Yasui, according to which the local metric admitting a non-degenerate rank 2 KY tensor must fall into the Carter family. It turns out that this is closely related to the fact that the self-dual PD solution preserves half of supersymmetry, whereas the non-self-dual solution admits only one quarter of supersymmetry. I will also explore the underlying mathematical structure of Euclidean PD solution, such as conformal ambi-Kahler structure and new type of Einstein-Weyl space.