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.
