Algebraic Criteria of the Integral Separation for Solving Certain Classes of Ordinary Differential Equations Systems

One of the important directions of the qualitative theory of ordinary differential equations is to study the properties of linear systems that satisfy the condition of integral separation. Anyway, integral separation becomes apparent in all studies concerning the asymptotic behavior of the solutions for the linear systems under the action of small perturbations. The papers of V.M. Millionschikov, B.F. Bylov, N.A. Izobov, I.N. Sergeev et al. proved that the available integral separation is the main reason for the rough stability of the characteristic Lyapunov exponents, the rough stability of the highest Lyapunov exponent, and the rough diagonalizability of systems by Lyapunov transformations, and other fundamental properties of linear differential systems. The paper presents the basic properties of the set of linear systems with constant, periodic, reducible coefficients and proves the algebraic criteria for their property of integral separation of solutions to be available. The results can be used in modeling dynamic processes.