  The original Jaynes-Cummings model is described by a Hamiltonian which is
exactly solvable. Here we extend this model by several types of interactions
leading to a nonhermitian operator which doesn't satisfy the physical condition
of space-time reflection symmetry (PT symmetry). However the new Hamiltonians
are either exactly solvable admitting an entirely real spectrum or quasi
exactly solvable with a real algebraic part of their spectrum.
