  We classify all surfaces with constant Gaussian curvature $K$ in Euclidean
$3$-space that can be expressed as an implicit equation of type
$f(x)+g(y)+h(z)=0$, where $f$, $g$ and $h$ are real functions of one variable.
If $K=0$, we prove that the surface is a surface of revolution, a cylindrical
surface or a conical surface, obtaining explicit parametrizations of such
surfaces. If $K\not=0$, we prove that the surface is a surface of revolution.
