The switched afine system considered in the paper comes to existence when stabilizing the chain of three integrators subject to the additional condition of asymptotic tracking a desired trajectory. The target trajectory is defined implicitly as that of the second-order integrator stabilized by means of a feedback in the form of nested saturators. The control law for the third-order integrator is suggested that ensures the fulfillment of the additional condition, and the range of parameters of the proposed control guaranteeing global stability of the closed-loop system is determined. Moreover, the problem of constructing ellipsoidal estimates of invariant sets of the system is stated and solved. In particular, we consider the problem of finding the best (in a certain sense) ellipsoidal estimate that guarantees that the deviation of the system from the equilibrium point at any time does not exceed a prescribed value as long as the initial state vector falls into the ellipsoid.