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A consequence of the Riemannian Goldberg-Sachs theorem is the fact that the Kähler form of an Einstein Hermitian surface is an eigenform of the curvature operator. Referring to this property as we obtain a complete classification of the compact locally homogeneous $*$-Einstein Hermitian surfaces. We also provide large families of non-homogeneous $*$-Einstein (but non-Einstein) Hermitian metrics on $ℂ{\mathbb{P}}^2♯{\overline{ℂ\mathbb{P}}}^2$, $ℂ{\mathbb{P}}^1\times ℂ{\mathbb{P}}^1$, and on the product surface $X\times Y$ of two curves $X$ and $Y$ whose genuses are greater than 1 and 0, respectively....

We provide a local classification of selfdual Einstein riemannian four-manifolds admitting a positively oriented hermitian structure and characterize those which carry a hyperhermitian, non-hyperkähler structure compatible with the negative orientation. We show that selfdual Einstein 4-manifolds obtained as quaternionic quotients of $ℍP^2$ and $ℍH^2$ are hermitian.