In this paper, we consider the following system of differential equations: è = oj + G(6,z), ¿ = Az + f(6,z), where 6 e Cm , w = {utx, ... , wm) e Rm , z e C" , A is a diagonalizable matrix, / and 0 are analytic functions in both variables and 27t-periodic in each component of the vector 6 , © = 0(\z\) and / = 0(|z|2) as z -> 0. We study the normal form of this system of the equations and prove that this system can be transformed to a system of linear equations 6 = a>, z = Az by an analytic

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... formation provided that the eigenvalues of A and the frequency co satisfy certain small-divisor conditions.