Qualitative methods of the Kolmogorov-Arnold-Mozer theory are used for investigation of a quasilinear oscillating system with the finite-dimensional frequency basis. A problem is formulated and studied on the existence of quasiperiodic solutions of a disturbed system with the same frequency basis using corresponding assumptions about arithmetic properties of characteristic indices of a generating system. The results of N.N. Bogolyubov and Ya.A. Mitropol'skij are generalized to the case when linear system matrix is not degenerated and allows purely imaginary eigenvalues. Along with consideration of so called algebraic critical case connected with investigations of the above mentioned authors a transcendental case of the matrix critical part is studied with corresponding Mozer's results being substantiated and refined.