On toric self-dual Einstein gravitational instantons

We consider the classification of toric self-dual Einstein gravitational instantons with negative cosmological constant. As is well known, any Killing vector field on a self-dual Einstein manifold defines a local conformal Kähler structure. We prove that if the conformal Kähler structure associated to one of the torus Killing fields is global and extends to an ALE manifold with no additional fixed points, then the corresponding self-dual Einstein instanton is precisely given by the infinite class of multipole solutions constructed by Calderbank, Pedersen and Singer.