Canonical metrics and ambiK\"ahler structures on 4-manifolds with $U(2)$ symmetry

For $U(2)$-invariant 4-metrics we express all curvature quantities up through the Bach tensor using an efficient computation framework. We fully classify the usual canonical metrics (Bach-flat, Einstein, extremal K\"ahler, etc) and also the $B^t$-flat metrics. Every $U(2)$-invariant metric is conformally K\"ahler in two separate ways, leading to ambiK\"ahler structures. Starting from the Taub-NUT and Taub-bolt metrics, we use ambiK\"ahler methods to find new complete extremal K\"ahler metrics on $\mathbb{C}^2$ and the total spaces of $\mathcal{O}(-1)$ and $\mathcal{O}(+1)$.