  We show that ergodic affine abelian discrete actions on the torus, that have
a rank-one factor in their linear part, are locally rigid in a KAM sense if and
only if the rank one factor is trivial and the action is higher-rank
transversally to this factor. Since it has been proved by Damjanovic and Katok
that affine actions with higher-rank linear part are locally rigid, our result
completes the local rigidity picture for affine actions on the torus.
