Аннотация:The author considers the system μx ' =A(t)x+h(t), x(0,μ)=x 0 . The matrix A(t) and the vector function h(t) are given and continuous in [0,T]. X is an μ-dimensional vector and μ is a small parameter. In a previous publication, the author imposed more restrictions on A(t), h(t) and the spectrum of A(t), and constructed an asymptotic integration algorithm for this system. In this paper with a variable t=τμ to reduce the above system to x ' =A(τμ)+h(μ), and the Jordan form of the matrix A(t), the author uses his earlier methods, hence assuming the same restrictions on A(t) and h(t) as in earlier publications, to construct an asymptotic expansion to the solution inside the subinterval [0,β ] where β is the width of the boundary layer. This boundary layer width is based on certain norms given in this paper. The author, with help from his earlier results, develops an iterative algorithm for his asymptotic integration. Through this algorithm a linearization of this system near the solution of the degenerate system is performed.