In this paper we are studying the problem of linear stationary singularly perturbed systems by the method of integral manifolds. The method of integral manifolds was proposed by M.M. Bogolyubov and J.O. Metropolsky for the study of perturbed nonlinear systems. This method was especially effective in the study of singularly perturbed systems of differential equations.

The first studies of integral manifolds and bounded solutions of linear singularly perturbed systems of differential equations were carried out in the works of Y.S. Baris and V.I. Fodchuk. The qualitatively new results in the study of singularly perturbed problems by the method of integral manifolds were obtained by V.A. Sobolev. Using integral manifolds of fast and slow variables, a variable replacement is built that transforms the input system to two independent subsystems. The results of V.A. Sobolev were generalized in the works of I.M. Cherevko and O.V. Osypova for the case of linear singularly perturbed systems with several small parameters.

The purpose of this work is to research methods of investigation of linear singularly perturbed stationary systems. An explicit form of non-degenerate replacement of variables is obtained, which splits the input system into two independent subsystems. The initial conditions were split and the principle of construction was established to study the stability of the zero solution. The possibility of using a zero approximation of the integral manifold of slow variables for investigating the stability of the solution of the input singularly perturbed system is considered.

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