In the theory of nonlinear oscillations one often encounters conditionally periodic oscillations resulting from the superposition of several oscillations with frequencies incommensurable with each other. When finding a solution to a resonant quasilinear differential system in the form of a conditionally periodic function, the problem of a small denominator arises. Consequently, the proof of the existence and even more the construction of such a solution is not an easy task. In this article, drawing on the work of V.I. Arnold, I. Moser, and other researchers proved the existence and constructed a conditionally periodic solution of a second-order quasilinear differential system in the critical case. Accelerated convergence method by N.N. Bogolyubova, Yu.A. Mitropolsky, A.M. Samoylenko. The result can be applied to construct a conditionally periodic solution of specific differential systems.