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We start discussing basic properties of Lie groupoids and Lie pseudo-groupsin view of applying these techniques to the analysis of Jordan-H\"olderresolutions and the subsequent integration of partial differential equationswhich is the summit of Lie and Cartan's work. Next, we discuss the integrationproblem for systems of partial differential equations in one unknown functionand special attention is given to the first order systems. The Grassmanniancontact structures are the basic setting for our discussion and the major partof our considerations inquires on the nature of the Cauchy characteristics inview of obtaining the necessary criteria that assure the existence ofsolutions. In all the practical applications of partial differential equations,what is mostly needed and what is in fact hardest to obtains are the solutionsof the system or, occasionally, some specific solutions. We continue ourdiscussion by examining the local equivalence problem for partial differentialequations, illustrating it with some examples, since almost any integrationprocess or method is actually a local equivalence problem involving a suitablemodel. We terminate the discussion by inquiring on non-integrable Pfaffiansystems and their integral manifolds of maximal dimension. This work is basedon four most enlightening M\'emoires written by \'Elie Cartan in the beginningof the last century.

On the Lie and Cartan Theory of Invariant Differential Systems, II