  In this paper, we study a local rigidity property of $\mathbb Z
\ltimes_\lambda \mathbb R$ affine action on tori generated by an irreducible
toral automorphism and a linear flow along an eigenspace. Such an action
exhibits a weak version of local rigidity, i.e., any smooth perturbations close
enough to an affine action is smoothly conjugate to the affine action up to
proportional time change.
