  We investigate the interplay between invariant varieties of vector fields and
the inflection locus of linear systems with respect to the vector field. Among
the consequences of such investigation we obtain a computational criteria for
the existence of rational first integrals of a given degree, bounds for the
number of first integrals on families of vector fields and a generalization of
Darboux's criteria. We also provide a new proof of Gomez-Mont's result on
foliations with all leaves algebraic.
