  This paper is devoted to the characterization of differentially flat
nonlinear systems in implicit representation, after elimination of the input
variables, in the differential geometric framework of manifolds of jets of
infinite order. We extend the notion of Lie-B\"acklund equivalence, introduced
in Fliess et al. (1999), to this implicit context and focus attention on
Lie-B\"acklund isomorphisms associated to flat systems, called trivializations.
They can be locally characterized in terms of polynomial matrices of the
indeterminate $\ddt$, whose range is equal to the kernel of the polynomial
matrix associated to the implicit variational system. Such polynomial matrices
are useful to compute the ideal of differential forms generated by the
differentials of all possible trivializations. We introduce the notion of a
strongly closed ideal of differential forms, and prove that flatness is
equivalent to the strong closedness of the latter ideal, which, in turn, is
equivalent to the existence of solutions of the so-called generalized moving
frame structure equations. Two sequential procedures to effectively compute
flat outputs are deduced and various examples and consequences are presented.
