Method of general Coule–Hopf substitutions in theory of finite-dimensional dynamical systems

We consider the results of applying the method of generic Cole–Hopf substitutions to integration of finite-dimensional dynamical systems. Dynamical systems are represented in the form of matrix ordinary differential equations with specific matrix algebra of finite dimension. The Cole–Hopf type substitutions are applied to matrix equations by using the differentiation on algebra in the form of commutator with a specific algebra element. Recurrent relations for Cole–Hopf substitutions were found. Particular cases of exactly integrable dynamical systems are presented. The algorithm of calculating the integrals of motion is shown.