  We first discuss the problems in the theory of ordinary differential
equations that gave rise to the concept of a flag system and illustrate these
with the Cartan criterion for Monge equations (1st order) as well as the Cartan
statement concerning the local equivalence of Monge-Amp\`ere type equations
(2nd order). Next, we describe a prolongation functor operating on the
infinitesimal symmetries (automorphisms) of the Darboux flag and extending
these, isomorphically, to all the symmetries of any other flag. Hence, flag
systems cannot be distinguished by their symmetry algebras and the local
classification of these objects is approached by considering higher order
isotropies of these algebras as well as the groupoids of $k-th$ order formal
equivalences since the differential equations defining the latter provide
precious information for the application of flag systems to differential
equations (\textit{e.g.}, Cartan's criterion for non-linear Monge equations).
In examining the behaviour of the isotropy algebras, that can either diminish
or remain the same, when passing from a derived system $S_\nu$ to the previous
system $S_{\nu-1}$, we obtain a full set of numerical invariants for the
elementary flag systems that moreover specify the local models.
