Quasi linear flows on tori: regularity of their linearization

Under suitable conditions a flow on a torus $C^{(p)}$--close, with $p$ large enough, to a quasi periodic diophantine rotation is shown to be conjugated to the quasi periodic rotation by a map that is analytic in the perturbation size. This result is parallel to Moser's theorem stating conjugability in class $C^{(p')}$ for some $p'<p$. The extra conditions restrict the class of perturbations that are allowed.