The loss of feedback linearizability of control systems is studied when restricted to a submanifold of the state-space. The slow approximation of singularly perturbed systems also falls into this category. We show that even if the static-feedback linearizability is lost, the flatness property of the system may be conserved. The comparison between a flexibly linked pendulum and a rigidly linked one, both equipped with two controls and evolving in the plane, illustrates the theoretical results.