Abstract. The following scheme proved to be the most effective in the analysis and solvability of nonlinear equations. An optimal generating equation (the equation of first approximation) is constructed by an optimal smoothing operator. The initial iteration is defined by this generating equation. Then, the generalized Krylov-Bogolyubov equation is used to determine higher iterations. In this method the error of iterations does not depend on the error of the initial approximation, whereas in classic methods this is not true. This is associated with the fact that, in the former, a sequence of transformations of phase spaces is performed, and, for a given problem, an optimal phase space is found. By the methods of computer algebra, one can construct in the analytic form an asymptotic solution to a nonlinear resonant system of differential equations whose right-hand sides are multiple Fourier series.