We numerically investigate the magneto-optical Cotton-Mouton effect in an alteating multilayer structure with a nonlinear dielectric constant. The multistability and polarization of the transmission of electromagnetic field near the edges of the stop gap are studied in detail. The resonant transmission is accompanied by solitons of intensity of the field. This investigation provides a way to select the transmission property with different polarizations since both the amplitude and the phase of the output field can be adjusted by the input power and by the magneto-optical coefficient depending on the exteal magnetic field.