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

Abstract

The purpose of this 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 the rank two prolongation. Utilizing this prolongation, we characterize the type of overdetermined systems and clarify the specificity for each type. We also give systematic methods for constructing the geometric singular solutions by analyzing a decomposition of this prolongation. As an application, we determine the geometric singular solutions of Cartan’s overdetermined system.