  A divergence-free vector field satisfies the star property if any
divergence-free vector field in some C1-neighborhood has all singularities and
all periodic orbits hyperbolic. In this paper we prove that any divergence-free
vector field defined on a Riemannian manifold and satisfying the star property
is Anosov. It is also shown that a C1-structurally stable divergencefree vector
field can be approximated by an Anosov divergence-free vector field. Moreover,
we prove that any divergence-free vector field can be C1-approximated by an
Anosov divergence-free vector field, or else by a divergence-free vector field
exhibiting a heterodimensional cycle.
