Relations among Hamiltonian, area-preserving, and non-wandering flows on surfaces

This paper gives a topological characterization of Hamiltonian flows with finitely many singular points on compact surfaces, using the concept of ``demi-caractéristique'' in the sense of Poincaré. Furthermore, we describe the relationships and distinctions among the Hamiltonian, divergence-free, and non-wandering properties for continuous flows, which gives an affirmative answer to the problem posed by Nikolaev and Zhuzhoma under the assumption of finitely many singular points.