The behaviour with respect to the set of variables ( x, t) of the one-dimensional manifold of solutions, tending to zero at infinity, of the set of two non-linear equations t −rx′ = A (t)x + ƒ (t, x), t 0 ⩽ t < ∞ , was investigated in [1], The results obtained enabled us to propose a method for solving the boundary value problem for such a set of equations in the interval [ t 0, ∞) with any given condition at the point t 0 and the limiting condition x( t) → 0 as t → ∞ at the right-hand end. The method consists in replacing [ t 0, ∞) by an interval [ t 0, T], taking as the boundary condition at the point T the equation of the entire stable manifold, and solving the boundary value problem in this finite interval. The present paper extends the results of [1] to a set of n equations. The condition x( t) → 0 as t → ∞ is removed from infinity to a finite point by means of the equation of the k-dimensional stable manifold, effectively defined by utilizing a convergent and asymptotic series. This type of sweep is justified here under the assumption that all k eigenvalues of the matrix A(∞), lying in the left-hand half-plane, are simple. We use the following notation: x = colon ( x i , …, x n ) is a real or complex vector column, diag ( D i , …, D s ) is a quasi-diagonal matrix, Ω x(c) is a closed region of the x-space, ¦x j¦⩽ c , j = 1, …, n; I t is a real semiinfinite interval T ⩽ t < ∞; Ω x, y(c) is the Cartesian product of the sets Ω x(c) × Ω y(c); ω x, t(c, T) = ω x(c) × I t ; ƒχƒ′ = ∑ j=1 n ƒχjƒ, where χ = (χ,…,χ n); ƒAƒ = ∑ i,j,=1 n ƒa ijƒ, where A = ( a i j ) is an n-th order square matrix, E = ( δ i j ) is the unit matrix, and [ x] m is a vector whose components are power series in the coordinates x, the powers of whose initial terms are not less than m. In addition, let l = ( l i , …, 1 n ) be a row-vector of non-negative integers, and x = colon ( x i , …, x n ). We take x l = x l 1 … x n l n , by definition, and define the length ¦l¦ of the vector l by the equation ƒlƒ = ∑ j=1 n lj . Then, for example, the series S = ∑ l 1…,l n=0 ∞ a l 1 … l n (t)χ l t 1…χ l n n ∑ j=1 n l j⩾1 may be written in the form S = ∑ ƒlƒ⩾1 a l(t)χ l . Throughout, we take T to be a large positive constant.