Drawing on results of Derdziński’s from the 1980s, we classify conformally Kähler, U(2)-invariant, Einstein metrics on the total space of \({{\mathcal {O}}}(-m)\) , for all \(m \in \mathbb {N}\) . This yields infinitely many 1-parameter families of metrics exhibiting several different behaviours including asymptotically hyperbolic metrics (more specifically of Poincaré type), ALF metrics, and metrics which compactify to a Hirzebruch surface \(\mathbb {H}_m\) with a cone singularity along the “divisor at infinity”. This allows us to investigate transitions between behaviours yielding interesting results. For instance, we show that a Ricci–flat ALF metric known as the Taub-bolt metric can be obtained as the limit of a family of cone angle Einstein metrics on \({\mathbb{C}\mathbb{P}}^2 \# \overline{{\mathbb{C}\mathbb{P}}}^2\) when the cone angle converges to zero. We also construct Einstein metrics which are asymptotically hyperbolic and conformal to a scalar-flat Kähler metric. Such metrics cannot be obtained by applying Derdziński’s theorem.
