On prolongations of second-order regular overdetermined systems with two independent and one dependent variables (Q2357390)

The author studies overdetermined systems consisting of two second-order partial differential equations for a scalar unknown function of two independent variables using a geometric approach based on jets and contact geometry. It is always assumed that the given system is regular in the sense that the restricted contact system has everywhere constant rank. The main purpose of the article is an investigation when there exist additional integral elements which are not transversal to the canonical projection. The set of all two-dimensional integral elements is called the rank two prolongation; the subset of those which are transversal the prolongation with transversality condition. Based on the structure equations of the symbol algebra, there already exists a classification of all regular systems of the given form into four different types called (I)--(IV). The first main result asserts that the two considered prolongations coincide, if and only if a system of type (II) or (III) is given. The remainder of the article is concerned with a deeper analysis of the type (I) case. It is first shown that the \(k\)th rank two prolongation defines an \(S^1\)-bundle over the \(k-1\)st. Then two different methods are discussed for the construction of geometrically singular solutions and demonstrated for a concrete system.