Abstract We show the following dichotomy for a linear parabolic Z^2 Z 2 -action ₋ ρ L on the torus with at least one step-2 generator: (i) Any affine Z^2 Z 2 -action with linear part ₋ ρ L has a ℤ-factor that is either identity or genuinely parabolic, and is thus not KAM-rigid, or (ii) Almost every affine Z^2 Z 2 -action with linear part ₋ ρ L is KAM-rigid under volume preserving perturbations.
Damjanović et al. (Mon,) studied this question.