Integrability by quadratures for systems of involutive vector fields

Abstract

Starting from results and ideas of S. Lie and E. Cartan, we give a systematic and geometric treatment of integrability by quadratures of involutive systems of vector fields, showing how a generalization of the usual multiplier can be constructed with the aid of closed differential forms and enough symmetry vector fields. This leads us to explicit formulas for the independent integrals. These results allow us to identify symmetries with integral invariants in the sense of Poincaré and Cartan. A further (new) result gives the equivalence of integrability by quadratures and the existence of solvable structures, these latter being generalizations of solvable algebras.