In this article, we study surfaces of revolution without parabolic points in a Euclidean 3-space whose Gauss map G satisfies the condition ${Delta}^hG;=;AG,A;{in};Mat(3,{mathbb{R}}),;where;{Delta}^h$ denotes the Laplace operator of the second fundamental form h of the surface and Mat(3,$mathbb{R}$) the set of 3${ imes}$3-real matrices, and also obtain the complete classification theorem for those. In particular, we have a characterization of an ordinary sphere in terms of it.