Minimal geodesic foliation on T^2 in case of vanishing topological entropy

On a Riemannian 2-torus $(T^2,g)$ we study the geodesic flow in the case of low complexity described by zero topological entropy. We show that this assumption implies a nearly integrable behavior. In our previous paper \cite{GK} we already obtained that the asymptotic direction and therefore also the rotation number exists for all geodesics. In this paper we show that for all $r\in \mathbb{R}\cup \{\mathrm{\infty }\}$ the universal cover ${\text{Br}}^2$ is foliated by minimal geodesics of rotation number $r$. For irrational $r\in \mathbb{R}$ all geodesics are minimal, for rational $r\in \mathbb{R}\cup \{\mathrm{\infty }\}$ all geodesics stay in strips between neighboring minimal axes. In such a strip the minimal geodesics are asymptotic to the neighboring minimal axes and generate two foliations.