We consider the problem of the existence of a solution of a two-point boundary-value problem for degenerate singularly perturbed linear systems of differential equations. We obtain asymptotic formulas for this solution.Consider the boundary-value problemwhere x.t; "/ is the required n-dimensional vector, t 2 OE0I 1; " 2 .0I " 0 / is a small parameter, h 2 N; A.t; "/ and B.t / are n n matrices, d is a p-dimensional column vector, f .t; "/ is an n-dimensional column vector, and M and N are p n matrices with constant elements. Assume that the following conditions are satisfied:(i) on the segment OE0I 1; the matrix A.t; "/ and vector f .t; "/ admit uniform asymptotic expansions in powers of the small parameter "; i.e.,(ii) the coefficients A k .t / and f k .t / of expansions (3) and the matrix B.t/ are infinitely differentiable on the segment OE0I 1I(iii) det B.t / D 0 8t 2 OE0I 1:Boundary-value problems of the type (1), (2) were investigated in [1][2][3][4][5] in the case where B.t/ D E; where E is the identity matrix. In [1,2], for the construction of asymptotic solutions of these problems, the method of regularization was used. In [3,4], the method of boundary functions was used for this purpose. In [5], the asymptotic behavior of a solution of a boundary-value problem was described using the idea of reduction of a linear system to an almost diagonal form. In [6], for the solution of this problem, the system was reduced to a system with a singular matrix coefficient of the derivative. In [7], a method was proposed for the asymptotic Nizhyn University, Nizhyn, Ukraine.

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