A new approach to solving the Boltzmann equation is based on the invariance of the collision integral with respect to the choice of basis functions. It is shown that the matrix elements corresponding to the moments of the nonlinear collision integral are related by simple recursive relations. All nonlinear matrix elements can be calculated if linear isotropic matrix elements are known. In isotropic case these relations were used to construct a numerical calculation procedure for power potentials which enables constructing the distribution function up to ten thermal velocities. The generalization of the obtained results on arbitrary potentials of interaction and on mixtures of gases is carried out.