Global Gevrey hypoellipticity on the torus for a class of systems of complex vector fields

Let $L_j={\mathrm{\partial }}_{t_j}+(a_j+i{b}_j)(t_j){\mathrm{\partial }}_x,j=1,\dots ,n,$ 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).