  We prove the existence of complete cohomogeneity one triaxial
K\"ahler-Einstein metrics in dimension four under an action of the Euclidean
group $E(2)$. We also demonstrate local existence of Ricci flat K\"ahler
metrics of a related type that are given via generalized PDEs, and determine,
under mild conditions, whether they are complete. The common framework for both
metric types is a frame-dependent system of Lie bracket relations and
generalized PDEs yielding a class of K\"ahler-Einstein metrics on $4$-manifolds
which includes all diagonal Bianchi type A metrics.
