  In this paper we study flows $\varphi:M\times\mathbb{R}\longrightarrow M$
having an isolated non-saddle set. We see that the complexity of the region of
influence of an isolated non-saddle set $K$ depends on the way in which $K$
sits on the phase space at the cohomological level. We construct flows in
surfaces having isolated non-saddle sets with a prescribed structure for its
region of influence. We also study parametrized families of smooth flows and
continuations of non-saddle sets.
