Darboux and analytic integrability of five-dimensional semiclassical Jaynes-Cummings system

We study the dynamics of a version of the Jaynes-Cummings system without the rotating wave approximation, which essentially consists of the interaction of two systems, n$$ n $$ two-level atoms and a single system of electromagnetic field. In particular, we investigate some types of first integrals such as Darboux, analytic, and time-dependent first integrals of this system. We prove that only two Darboux first integrals and two analytic first integrals exist for this system. Moreover, we determine all exponential factors and invariant algebraic hypersurfaces of this system.