We obtain in this work, in an intrinsic way, the First classification of Pfaffian systems of rank two and three in dimension five, as well as a complete classification up to dimension four. For the general case, we prove that there exists a one-to-one correspondence, which is explicitly given, between Pfaffian systems of rank two and Pfaffian systems of rank three. That is, for a Pfaffian system of rank two in dimension five, the covariant system is constructed.