Normal Singular Geodesics of a Conformally Generic Sub-Riemannian Metric

We prove that a Mañé generic real-analytic $D$-Hamiltonian H, subjected to a totally non-holonomic real-analytic distribution $D$, has no non-trivial normal $D$-singular orbits of minimal rank. If $D$ has co-rank 1, this implies that $H+u$, where $u$ is a generic real-analytic potential, does not admit non-trivial normal $D$-singular orbits.