For an Anosov dynamical system on the three torus there are three invariant vector fields associated with the Lyapunov exponents. Each of these vector fields generates an invariant foliation of the torus. In the case of a linear automorphism, these foliations have dense leaves. It is unclear under which condition this property is valid for a more general system. For a smooth perturbation of a linear automorphism it is possible to explicitly construct, via a simple perturbative expansion,  the invariant vector fields together with the conjugation between the perturbed map and the unperturbed one. One can naively expect that the image, under the conjugation, of the invariant foliations for the unperturbed system gives the invariant foliations for the perturbed one. This is generally not the case. Using the explicit perturbative expansions for these objects we will try to shed some light on their relations.