The qualitative methods of the Kolmogorov-Arnol'd-Moser theory are used to investigate a quasilinear oscillatory system with finite-dimensional frequency basis. The question of whether a perturbed system with the same basis has quasiperiodic solutions is formulated and studied, subject to suitable assumptions concerning the arithmetical properties of the characteristic indices of the generating system. The Bogolyubov-Mitropol'skii results are extended to the case in which the matrix of the linear system is non-singular and has purely imaginary eigenvalues. The existence of integral manifolds of a certain type is proved using the structural properties of the system, by almost identical transformations of the variables, the small parameter being scaled in a power sense. Besides the so-called algebraic critical case associated with the above-mentioned author's work, some attention is devoted to the transcendental case of the critical part of the matrix, at the same time justifying and sharpening Moser's results in this area.