KAM-rigidity for parabolic affine abelian actions

We show the following dichotomy for a linear parabolic ${\mathbb{Z}}^2$-action ${\rho }_L$ on the torus with at least one step-2 generator: (i) Any affine ${\mathbb{Z}}^2$-action with linear part ${\rho }_L$ has a $\mathbb{Z}$-factor that is either identity or genuinely parabolic, and is thus not KAM-rigid, or (ii) Almost every affine ${\mathbb{Z}}^2$-action with linear part ${\rho }_L$ is KAM-rigid under volume preserving perturbations.