  In this article, we study the class of surfaces of revolution in the
3-dimensional Euclidean space $E^{3}$ with nonvanishing Gauss curvature whose
position vector $\boldsymbol{x}$ satisfies the condition
$\Delta^{II}\boldsymbol{x}=A\boldsymbol{x}$, where $A$ is a square matrix of
order 3 and $\Delta^{II}$ denotes the Laplace operator of the second
fundamental form $II$ of the surface. We show that a surface of revolution
satisfying the preceding relation is a catenoid or part of a sphere.
