  The purpose of this paper is to study immersed surfaces in the product spaces
$\mathbb{M}^2(\kappa)\times\mathbb{R}$, whose mean curvature is given as a
$C^1$ function depending on their angle function. This class of surfaces
extends widely, among others, the well-known theory of surfaces with constant
mean curvature. In this paper we give necessary and sufficient conditions for
the existence of prescribed mean curvature spheres, and we describe complete
surfaces of revolution proving that they behave as the Delaunay surfaces of CMC
type.
