On solutions of a quasilinear differential second-order system represented by Fourier series with slowly varying parameters in some critical cases

We study a quasilinear second-order differential system whose coefficients are represented in the form of absolutely and uniformly convergent Fourier series with slowly varying coefficients and the frequency. The conditions of existence of a particular solution of a similar structure are obtained in the case where the eigenvalues of the matrix of the linear part of the system are equal to zero.