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We are concerned with the systems: (1) y = A(t)y (2) x + A(t)x + f(t,x) where x, y and f(t,x) are n-vectors, A(t) is a n x n matrix and t is in J = [t_o,∞), t_o ≥ 0. We give conditions on f(t,x) and A(t) in such a way that the following statements hold: (I) If y(t) ≠ 0 is a solution of (1) then there exist a family of solutions x(t) of (2) satisfying (3) . lim_t→∞ ΙΙx(t)-y(t)ΙΙ / Ιιy(t)ΙΙ =0 and ΙΙx(t)-y(t)ΙΙ / ΙΙy(t)ΙΙ integrable on J (II) If x(t) is a solution of (2), x(t) ≠ 0 for all large t, then there exist a family of solutions y(t) of (2) such that (3) and (4) hold.