  In this article, we study the class of surfaces of revolution in the
3-dimensional Lorentz-Minkowski space with nonvanishing Gauss curvature whose
position vector x satisfies the condition {\Delta}IIIx = Ax, where A is a
square matrix of order 3 and {\Delta}III denotes the Laplace operator of the
second fundamental form III of the surface. We show that such surfaces are
either minimal or pseudospheres of a real or imaginary radius.
