The nonequilibrium statistical operator method is used to develop a theory of combined resonance in semiconducting crystals with the CdS lattice that is nonlinear in an external fie ld. A system of stationary energy balance equations is set up and solved for the kinetic and spin degrees of freedom of conduction electrons interacting with nuclear spins in the nonlinear regime of combined resonance. An analysis of the solutions reveals that new schemes of dynamical polarization of nuclei arise under these conditions. 1. In the earlier paper [1] we constructed an effective interaction of conduction electrons with an external electromagnetic field under conditions typical of combined resonance. This interaction determines gauge-invariant equations of motion of the macroscopic variables of the system. Here, we consider combined resonance in semiconducting crystals of the CdS type, in which a minimum of the energy of the conduction electrons is attained on a circle (loop of extrema) in the Brillouin zone with center on a symmetry axis. In crystals of this type, the operator of the interaction of the kinetic and spin degrees of freedom of the electrons is already nonvanishing in the linear approximation in the electron momentum [2]. This guarantees high intensity of the combined resonance. To obtain macroscopic balance equations describing the nonlinear regime of combined resonance in these crystals, we apply the nonequilibrium statistical operator method [3, 4] in conjunction with the gaugeinvariant interaction of [1]. The aim of our paper is to study the solutions of these equations in the region of quadratic nonlinearity in the field strengths. We shall show that in these crystals one can have new schemes of dynamical polarization of nuclear spins that interact with the nonequilibrium electrons in the nonlinear regime of combined resonance. Nuclei are polarized when one passes through all the resonance frequencies of the system corresponding to cyclotron, paramagnetic, and combined resonance; the polarization of largest absolute magnitude corresponds to saturation of combined resonance at an antinode of the electric component of the field. The direction of the nuclear polarization depends on the nature of the resonance transition. For example, for cyclotron and combined resonance in an alternating electric field the directions of polarization are opposed, whereas for combined resonance in an alternating electric field and paramagnetic resonance in alternating magnetic field the directions of the polarization coincide. The transfer of energy produced by the interaction of the subsystems is shown in Fig. 1. The energy f of the external electromagnetic field enters the system through the channels f -* k and f ~ s, into the subsystems of the kinetic (k) and spin (s) degrees of freedom of the electrons. The energy is then transferred to the lattice (l) both directly and through the nuclear spin subsystem (n). The dashed line shows the original interaction of the subsystems k and s responsible for the existence of the combined resonance; it is eliminated by a canonical transformation of the Hamiltonian. 2. We consider the hexagonal modification of the CdS type crystals in a magnetic field H along the hexagonal axis. The Hamiltonian of the conduction electrons and nuclear spins, which interact with each other and with the lattice, can be written in the form*