Minimal points and non-holonomic controllability on compact manifolds

We study the problem of non-holonomic point-to-point controllability for ODEs with drift possessing some recursion property of the flow (nonwandering or chain recurrence) and satisfying various versions of Hörmander condition (also known as Lie bracket generating condition). We show that for the flows on compact manifolds, it suffices to assume the validity of the Hörmander condition on the closure of the set of their minimal points only. Also, we construct a 2-dimensional example of a drift defining a chain recurrent flow and the vector fields defining the non-holonomic constraint, which together satisfy the Hörmander condition, but the flow is not controllable in the direction of the given vector fields.