Ricci-flat metrics on the cone over ${\mathbb{C}\mathbb{P}}^2\mathrm{\#}\overline{{\mathbb{C}\mathbb{P}}^2}$

We describe a framework for constructing the Ricci-flat metrics on the total space of the canonical bundle over ${\mathbb{C}\mathbb{P}}^2\mathrm{\#}\overline{{\mathbb{C}\mathbb{P}}^2}$ (the del Pezzo surface of rank one). We construct explicitly the first-order deformation of the so-called `orthotoric metric’ on this manifold. We also show that the deformation of the corresponding conformal Killing-Yano form does not exist.