  As we have proved in [L], the geodesic flows associated with the flat metrics
on T^2 minimize the polynomial entropy. In this paper, we show that, among the
geodesic flows that are Bott integrable and dynamically coherent, the geodesic
flows associated to flat metrics are local strict minima for the polynomial
entropy. To this aim, we prove a graph property for invariant Lagrangian tori
in near-integrable systems.
