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A control system $\dot{q} = f(q,u)$ is said to be trivializable if thereexists local coordinates in which the system is feedback equivalent to acontrol system of the form $\dot{q} = f(u)$. In this paper we characterizetrivializable control systems and control systems for which, up to a feedbacktransformation, $f$ and $\partial f/\partial u$ commute. Characterizations aregiven in terms of feedback invariants of the system (its control curvature andits centro-affine curvature) and thus are completely intrinsic. To conclude weapply the obtained results to Zermelo-like problems on Riemannian manifolds.

Control systems of zero curvature are not necessarily trivializable