  In [15], V. Jimenez and J. Llibre characterized, up to homeomorphism, the
omega limit sets of analytic vector fields on the sphere and the projective
plane. The authors also studied the same problem for open subsets of these
surfaces.
  Unfortunately, an essential lemma in their programme for general surfaces has
a gap. Although the proof of this lemma can be amended in the case of the
sphere, the plane, the projective plane and the projective plane minus one
point (and therefore the characterizations for these surfaces in [8] are
correct), the lemma is not generally true, see [15].
  Consequently, the topological characterization for analytic vector fields on
open subsets of the sphere and the projective plane is still pending. In this
paper, we close this problem in the case of open subsets of the sphere.
