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

KAM rigidity for partially hyperbolic affine Z^k actions on the torus with a rank one factor