Global Denjoy–Carleman hypoellipticity for a class of systems of complex vector fields and perturbations

We characterize global Denjoy-Carleman hypoellipticity for the family of systems acting on the torus T^{N}_{t} x T_{x} given by L_j = \partial_{t_j} + (a_j(t_j) ib_j(tj_j))\partial_x + lambda_j, where a_j, b_j are real-valued 2\pi-periodic elements of the classes and j = 1, 2,., N.