Qualitative properties of a certain kinetic model of a binary gas

We consider a system of kinetic equations with one-dimensional velocity space. The system is a simple mathematical model that describes the evolution of a two-component gas mixture at the molecular level. We study some qualitative properties of its solutions, in particular, the conservation laws and spectrum of the linearized problem. In the spatially homogeneous case we present the widest Lie algebra of admissible operators and construct some exact solutions in closed form. We indicate some methods for constructing numerical schemes conservative with respect to fulfillment of the discrete conservation laws of energy and the concentrations of the components.