An approach to the identification of non-linear maximally stationary systems is proposed, based on the use of a consistent measure of dependence of the input and output processes of the system under study. In accordance with the conventional terminology, a measure of dependence between two random values (processes) is referred to as consistent, if it vanishes if and only if the values (processes) are stochastically independent. Within the consideration subject, such a measure of dependence is the maximal correlation. In turn, the maximally stationary systems are those, for which the first eigenfunctions, corresponding to the largest in the absolute value first eigenvalue of the joint probability distribution density expansion, do not depend on the time.