  Area-preserving flows on compact surfaces are one of the classic examples of
dynamical systems, also known as multi-valued Hamiltonian flows. Though
Hamiltonian, area-preserving, and non-wandering properties for flows are
distinct, there are some equivalence relations among them in the
low-dimensional cases. In this paper, we describe equivalence and difference
for continuous flows among Hamiltonian, divergence-free, and non-wandering
properties topologically. More precisely, let $v$ be a continuous flow with
finitely many singular points on a compact surface. We show that $v$ is
Hamiltonian if and only if $v$ is a non-wandering flow without locally dense
orbits whose extended orbit space is a directed graph without directed cycles.
Moreover, non-wandering, area-preserving, and divergence-free properties for
$v$ are equivalent to each other.
