Geometric Methods to Investigate Prolongation Structures for Differential Systems with Applications to Integrable Systems

A type of prolongation structure for several general systems is discussed. They are based on a set of one forms for which the underlying structure group of the integrability condition corresponds to the Lie algebra of SL(2, R), O(3), and SU(3). Each will be considered in turn and the latter two systems represent larger 3 × 3 cases. This geometric approach is applied to all of the three of these systems to obtain prolongation structures explicitly. In both 3 × 3 cases, the prolongation structure is reduced to the situation of three smaller 2 × 2 problems.