Geometry of surfaces corresponding to O(3, 2)/O(2,1 X O(1,1) - system

In this paper we continue the study started by Terng C-L., of the geometry of the submanifolds associated to the $U/K$-system, for the particular case $U/K$ being the space $O(3, 2)/O(2,1)\times O(1,1)$. Following the ideas from Bruck-Du-Park-Terng, we conclude that these include flat time-like surfaces in pseudo-riemannian spaces and isothermic space-like surfaces in the Lorentzian space. We also obtain an explicit action of a rational map with two simple poles on the space of solutions of the $O(3, 2)/O(2,1)\times O(1,1)$-system, and conclude that these correspond to Darboux and Ribaucour transformations for space-like surfaces in pseudo-euclidean space $\mathbb R^{2,1}$.