Non-integrability of the semiclassical Jaynes-Cummings models without the rotating-wave approximation

Two versions of the semiclassical Jaynes-Cummings model without the rotating wave approximation are investigated. It is shown that for a non-zero value of the coupling constant the version introduced by Belobrov, Zaslavsky, and Tartakovsky is Hamiltonian with respect to a certain degenerated Poisson bracket. Moreover, it is proved that both models are not integrable.