  An integrable Hamiltonian system presents monodromy if the action-angle
variables cannot be defined globally. We consider a classical system with
azimuthal symmetry and explore the topology structure of its phase space. Based
on the behavior of closed orbits around singular points or regions of the
energy-momentum plane, a semi-theoretical method is derived to detect classical
monodromy. The validity of the monodromy test is numerically illustrated for
several systems with azimuthal symmetry.
