Existence of Birkhoff sections for Kupka-Smale geodesic flows of closed surfaces

We show that, on a closed surface, any Riemannian metric satisfying the Kupka-Smale condition admits a Birkhoff section for its geodesic flow. In particular, this implies that a $C^\infty$-generic Riemannian metric on a closed surface admits a Birkhoff section for its geodesic flow.