Abstract: In this paper, a second-order discrete-time automatic control system is studied. This study is a continuation of the research presented in the author’s papers “On the Aizerman problem: coefficient conditions for the existence of a four-period cycle in a second-order discrete-time system” and “On the Aizerman problem: coefficient conditions for the existence of three- and six-period cycles in a second-order discrete-time system,” where systems with two- and three periodic nonlinearities lying in the Hurwitz angle were considered. The systems with nonlinearities subjected to stronger constraints are discussed in this paper. It is assumed that the nonlinearity not only lies in the Hurwitz angle but also satisfies the additional sector condition. This formulation of the problem can be found in many works devoted to theoretical and applied problems of the automatic control theory. In this paper, a system with such a nonlinearity is investigated for all possible values of the parameters. It is shown that in this case, there are the parameter values for which a system with a two-periodic nonlinearity has a family of four-period cycles and a system with a three-periodic nonlinearity has a family of three- or six-period cycles. The conditions on the parameters under which the system can have a family of periodic solutions are written out explicitly. The proofs of the theorems provide a method for constructing a nonlinearity in such a way that any solution of the system with the initial data lying on some definite ray is periodic.