  We study the focus-focus type of monodromy in an integrable version of the
Dicke model. Classical orbits forming a pinched torus represent analogues of
the dynamic superradiance under conditions of a closed system. Quantum
signatures of monodromy appear in lattices of expectation values of various
quantities in the Hamiltonian eigenstates and are related to an excited-state
quantum phase transition. We demonstrate the breakdown of these structures with
an increasing strength of non-integrable perturbation.
