@BenMcKay Thank you very much for the reference :)

@YCor, the completeness issue can be bypassed: the 'geodesic' flow defines a foliation on T^1(S) which is a compact manifold, and all dynamical questions can be rephrased in terms of this foliation. If you really insist on rephrasing the question using flow formalism, you can build a smooth vector field on T^1(S) whose integral curves are the leaves of the aforementioned foliation.

Thank you very much Matheus for taking the time to give this detailed answer!

Oh yes, indeed.

You are perfectly right, once again my question is not precise enough; I must ask that the transformation I am considering is in 'reduced form' that is with as little continuity intervals as possible up to conjugating by a rotation.

Yes of course that's what I meant :)