Poincarè-Bendixson theory for parabolic holomorphic foliations by curves

The classical  Poincar\'e - Bendixson theory describes the way a trajectory of
a vector field on the real plane behaves when both accumulating, or not to the
singular locus of a vector field in question. We describe, in the first approximation,
the way a leaf with contractible holonomy of a parabolic holomorphic foliation by
curves on the complex manifold approaches the singular locus of the foliation.