The topology associated with cusp singular points

In this paper we investigate the global geometry associated with cusp singular points of two-degree of freedom completely integrable systems. It typically happens that such singular points appear in couples, connected by a curve of hyperbolic singular points. We show that such a couple gives rise to two possible topological types as base of the integrable torus bundle, that we call pleat and flap. When the topological type is a flap, the system can have non-trivial monodromy, and this is equivalent to the existence in phase space of a lens space compatible with the singular Lagrangian foliation associated to the completely integrable system.