Integrable Hamiltonian systems with swallowtails

We consider two-degree-of-freedom integrable Hamiltonian systems with bifurcation diagrams containing swallowtail structures. The global properties of the action coordinates in such systems together with the parallel transport of the period lattice and corresponding quantum cells in the joint spectrum are described in detail. The relation to the concept of bidromy which was introduced in SadovskiI and ZhilinskiI (2007 Ann. Phys. 322 164-200) is discussed.