Surfaces of coordinate finite $ II $-type

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.