Integrable dynamic chains and method of functional substitutions

Background. The paper outlines the main elements of the theory of functional substitutions for constructing integrable dynamical systems in the form of infinite and finite discrete chains of elements, similar to Toda chains, but not in the general case Hamiltonian. The aim of the work is to construct a general scheme for constructing the integrable equations themselves, their solutions, and integrals of motion. Method. The method of research is the method of functional substitutions developed earlier in the works of the authors in the form applicable to discrete dynamic systems in the form of infinite and finite chains of equations that are models of many models in physical and biological kinetics. Results. The paper developed a general scheme for applying the method of functional substitutions to the construction and integration of the equations of dynamics of infinite and finite chains of equations. A number of specific models and their common solutions are considered. An important result of the work is the construction of exact solutions of finite dynamical systems and their integrals of motion, which play an important role in physical and biological kinetics. Conclusions. It was shown that in addition to the known types of discrete chains of the type of Toda chains that are integrable using the method of the inverse problem, there are many chains that are integrable using the method of functional substitutions. These discrete chains can also be considered as useful models in various applications. An important conclusion is that the method of functional substitutions allows to obtain solutions of a set of models, the integrability of which was not previously known.