  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.
