### Weakly-Einstein hermitian surfaces

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 $\u2102{\mathbb{P}}^{2}\u266f{\stackrel{\u203e}{\u2102\mathbb{P}}}^{2}$, $\u2102{\mathbb{P}}^{1}\times \u2102{\mathbb{P}}^{1}$, and on the product surface $X\times Y$ of two curves $X$ and $Y$ whose genuses are greater than 1 and 0, respectively....