On prolongations of second-order regular overdetermined systems with two independent and one dependent variables

The purpose of this present paper is to investigate the geometric structure of regular overdetermined systems of second order with two independent and one dependent variables from the point of view of rank 2 prolongations. Utilizing this notion of prolongations, we characterize the type of these overdetermined systems. We also give a systematic method for constructing the geometric singular solutions by analyzing of a decomposition of this prolongation. As an applications, we determine the geometric singular solutions of Cartan's overdetermined systems.