  Let $(S,\Phi)$ be a pair of a closed oriented surface and $\Phi$ be a real
analytic flow with finitely many singularities. Let $x$ be a point of $S$ with
the polycycle $\omega$-limit set $\omega(x)$. In this paper we give topological
classification of $\omega(x)$. Our main theorem says that $\omega(x)$ is
diffeomorphic to the boundary of a cactus in the $2$-sphere $S^{2}$. Moreover
$S$ is a connected sum of the above $S^{2}$ and a closed oriented surface along
finitely many embedded circles which are disjoint from $\omega(x)$. This gives
a natural generalization to the higher genus of the main result of \cite{JL}
for the genus $0$ case. Our result is further applicable to a larger class of
surface flows, a compact oriented surface with corner and a $C^{1}$-flow with
finitely many singularities locally diffeomorphic to an analytic flow.
