On invariant manifolds of linear differential equations. I

Abstract: This article is the first in the cycle from two parts. It develops the ideas of integral manifolds method of M. M. Bogolubov in the case of linear differential equations in $R^m$ with variable coefficients. We distinguish linear subspaces $M^n(t)$ and $M^{m-n}(t)$, which have dimensions $n$ and $m-n$ respectively, $m > n$, such that $R^m = M^n(t) \oplus M^{m-n}(t)$, and find necessary and sufficient conditions under which these subspaces are invariant with respect to differential equation under consideration.