Anisotropic solutions of a nonlinear kinetic model of elliptic type

We consider a nonlinear kinetic model described by a system of two equations in partial derivatives of the elliptic type with exponential nonlinearities. We propose to construct exact solutions of the mathematical model in the class of logarithms from quadratic functions of spatial variables. The solution coefficients of the model are found from systems of square matrix and linear vector equations. In particular, the proposed approach is used to construct anisotropic solutions to the Liouville equation, often used as a mathematical model of stationary distributions in plasma physics. We illustrate the results by a number of examples.