We study the class of surfaces of revolution in the 3-dimensional Euclidean space $E^3$ with nonvanishing Gauss curvature whose position vector $\mathit{x}$ satisfies the condition ${\mathrm{\Delta }}^{II}\mathit{x}=A\mathit{x}$, where $A$ is a square matrix of order 3 and ${\mathrm{\Delta }}^{II}$ 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.

In this paper we studied quadric surfaces in the Euclidean 3-space that were of finite type with respect to the second fundamental form Ⅱ. The main result presented in this article was that spheres were the only quadric surfaces of finite type. This indicated a specific and notable classification within the broader category of quadric surfaces based on their finite type characteristics in relation to the second fundamental form.