Non-Integrable Pfaffian Systems

We discuss a recurrent geometrical method, due to \'Elie Cartan and von Weber ([1],[11]) enabling us to determine, step by step, the maximal integral manifolds of a not necessarily integrable nor regular Pfaffian system. The dimensions of such integral manifolds can, of course, vary from point to point but more so can vary at a given point it depending upon the choice of their recurrent buildup. When the system is regular and integrable then, of course, we obtain the maximal integral leaves of the integral foliation. Attention is also given to those integrable systems that can be integrated by quadratures which was, in the 19th century, the dream of many. However, our main interest resides in enhancing the Jordan-H\"older integration procedure so as to construct the local maximal integral manifolds that find many applications.