  Let $L_j = \partial_{t_j} + (a_j+ib_j)(t_j) \partial_x, \, j = 1, \dots, n,$
be a system of vector fields defined on the torus
$\mathbb{T}_t^{n}\times\mathbb{T}_x^1$, where the coefficients $a_j$ and $b_j$
are real-valued functions belonging to the Gevrey class $G^s(\mathbb{T}^1)$,
with $s>1$. In this paper we were able to characterize the global
$s-$hypoellipticity of this system in terms of Diophantine approximations and
the Nirenberg-Treves condition (P).
