  Given $a,b\in\mathbb{R}$ and $\Phi\in C^1(\mathbb{S}^2)$, we study immersed
oriented surfaces $\Sigma$ in the Euclidean 3-space $\mathbb{R}^3$ whose mean
curvature $H$ and Gauss curvature $K$ satisfy $2aH+bK=\Phi(N)$, where
$N:\Sigma\rightarrow\mathbb{S}^2$ is the Gauss map. This theory widely
generalize some of paramount importance such as the ones constant mean and
Gauss curvature surfaces, linear Weingarten surfaces and self-translating
solitons of the mean curvature flow. Under mild assumptions on the prescribed
function $\Phi$, we exhibit a classification result for rotational surfaces in
the case that the underlying fully nonlinear PDE that governs these surfaces is
elliptic or hyperbolic.
