[en] A model is presented to investigate the problem of interaction between an N-level atom and (N-1) modes of the field. The model includes the effect of detuning. Constants of motion are obtained. The evolution operator is calculated. The probability distribution function for the photons numbers is computed for different initial atomic states. The characteristic functions are computed. Different statistical quantities concerning the photons or the atomic system are given. It is found that if the atom starts from a definite state then the photon number never exceeds its initial value in the mode that effects the interaction between this state and the upper-most state. On the other hand, in the other modes the photon numbers at any time are never less than the initial value. (author)