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Drawing on results of Derdzi\'nski's from the 80's, we classify conformallyK\"ahler, $U(2)$-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 behavioursincluding asymptotically hyperbolic metrics (more specifically of Poincar\'etype), ALF metrics, and metrics which compactify to a Hirzebruch surface$\mathbb{H}_m$ with a cone singularity along the ''divisor at infinity''. Thisallows us to investigate transitions between behaviours yielding interestingresults. For instance, the recently discovered Ricci-flat ALF metric known asthe Chen-Teo instanton can be obtained as a limit of a family of cone angleEinstein metrics on $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$ when the coneangle converges to zero. We also construct Einstein metrics which areasymptotically hyperbolic and conformal to a scalar-flat K\"ahler metric. Suchmetrics cannot be obtained by applying Derdzi\'nski's theorem.

Einstein metrics from the Calabi ansatz via Derdzi\'nski duality