  We apply the original semiclassical approach to the kinetic ionization
equation with the nonlocal cubic nonlinearity in order to construct the family
of its asymptotic solutions. The approach proposed relies on an auxiliary
dynamical system of moments of the desired solution to the kinetic equation and
the associated linear partial differential equation. The family of asymptotic
solutions to the kinetic equation is constructed using the symmetry operators
acting on functions concentrated in a neighborhood of a point determined by the
dynamical system. Based on these solutions, we introduce the nonlinear
superposition principle for the nonlinear kinetic equation. Our formalism based
on the Maslov germ method is applied to the Cauchy problem for the specific
two-dimensional kinetic equation. The evolution of the ion distribution in the
kinetically enhanced metal vapor active medium is obtained as the nonlinear
superposition using the numerical-analytical calculations.
