Euclidean supersymmetric solutions with the self-dual Weyl tensor

We explore the Euclidean supersymmetric solutions admitting the self-dual gauge field in the framework of ${\cal N}=2$ minimal gauged supergravity in four dimensions. According to the classification scheme utilizing the spinorial geometry or the bilinears of Killing spinors, the general solution preserves one quarter of supersymmetry and is described by the Przanowski-Tod class with the self-dual Weyl tensor. We demonstrate that there exists an additional Killing spinor, provided the Przanowski-Tod metric admits a Killing vector that commutes with the principal one. The proof proceeds by recasting the metric into another Przanowski-Tod form. This formalism enables us to show that the self-dual Reissner-Nordstr\"om-Taub-NUT-AdS metric possesses a second Killing spinor, which has been missed over many years. We also address the supersymmetry when the Przanowski-Tod space is conformal to each of the self-dual ambi-toric K\"ahler metrics. It turns out that three classes of solutions are all reduced to the self-dual Carter family, by virtue of the nondegenerate Killing-Yano tensor.