Painlevé test and a self-similar solution of the kinetic model

We study a one-dimensional system of equations for a discrete gas model (McKean system). The McKean system is the Boltzmann kinetic equation of a model one-dimensional gas consisting of two groups of particles. Under certain conditions on a singularity variety, the system passes the Painlevé test. In addition, the kinetic system admits a reduction to a system of ordinary differential equations for which the Painlevé test is performed and it becomes possible to find a solution.