  We consider smooth Riemannian surfaces whose curvature $K$ satisfies the
relation $\Delta\log|K-c|=aK+b$ away from points where $K=c$ for some
$(a,b,c)\in\mathbb{R}^3$, which we call generalized Ricci surfaces. We prove
some isometric immersion theorems allowing points where $K=c$ using properties
of log-harmonic functions. For instance, we obtain a characterization of
Riemannian surfaces that locally admit minimal isometric immersions, possibly
with umbilical points, into a $3$-dimensional Riemannian manifold of constant
sectional curvature. We also give an application to convex affine spheres.
Finally, we study compact generalized Ricci surfaces: we obtain topological
obstructions and construct examples.
