  An integrable Hamiltonian system presents monodromy if the action-angle
variables cannot be defined globally. As a prototype of classical monodromy
with azimuthal symmetry, we consider a linear molecule interacting with
external fields 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.
