  We state some generalizations of a theorem due to G. Darboux, which
originally states that a polynomial vector field in the complex plane exhibits
a rational first integral and has all its orbits algebraic provided that it
exhibits infinitely many algebraic orbits. In this paper, we give an
interpretation of this result in terms of the classical Reeb stability
theorems, for compact leaves of (non-singular) smooth foliations. Then we give
versions of Darboux's theorem, assuring, for a (non-singular) holomorphic
foliation of any codimension, the existence of an open set of compact leaves
provided that the measure of the set of compact leaves is not zero. As for the
case of polynomial vector fields in the complex affine space of dimenion
$m\geq2$, we prove suitable versions of the above results, based also on the
very special geometry of the complex projective space of dimension $m$, and on
the nature of the singularities of such vector fields we consider.
