Abstract: In this paper, an automatic control discrete-time system of the second order is studied. The nonlinearity of this system satisfies the generalized Routh–Hurwitz condition. Systems of this type are widely used in solving modern applied problems of the theory of automatic control. This work is a continuation of the results of research presented in the paper “On the Problem of Aizerman: Coefficient Conditions for the Existence of a Four-Period Cycle in a Second-Order Discrete-Time System,” in which systems with two-periodic nonlinearity lying in the Hurwitz angle were studied. In above-mentioned paper, the conditions on the parameters under which a system with two-periodic nonlinearity can possess a family of nonisolated four-period cycles are indicated and a method for constructing such nonlinearity is proposed. In the current paper, we assume that the nonlinearity is three-periodic and lies in a Hurwitz angle. We study a system for all possible parameter values. We explicitly present the conditions for the parameters under which it is possible to construct a three-periodic nonlinearity in such a way that a system with specified nonlinearity is not globally asymptotically stable. We show that a family of three-period cycles and a family of six-period cycles can exist in the system with this nonlinearity. A method for constructing such nonlinearities is proposed. The cycles are nonisolated; any solution of the system with the initial data, which lies on a certain specified ray, is a periodic solution.