C∞-structures in the integration of involutive distributions

Abstract

For a system of ordinary differential equations (ODEs) or, more generally, an involutive distribution of vector fields, the problem of its integration is considered. Among the many approaches to this problem, solvable structures provide a systematic procedure of integration via Pfaffian equations that are integrable by quadratures. In this paper structures more general than solvable structures (named C∞-structures) are considered. The symmetry condition in the concept of solvable structure is weakened for C∞-structures by requiring their vector fields be just C∞-symmetries. For C∞structures there is also an integration procedure, but the corresponding Pfaffian equations, although completely integrable, are not necessarily integrable by quadratures. The well-known result on the relationship between integrating factors and Lie point symmetries for first-order ODEs is generalized for C∞-structures and involutive distributions of arbitrary corank by introducing symmetrizing factors. The role of these symmetrizing factors on the integrability by quadratures of the Pfaffian equations associated with the C∞-structure is also established. Some examples that show how these objects and results can be applied in practice are also presented.