  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 $r \in \mathbb{R} \cup \{\infty\}$ the universal cover $\Br^2$ is foliated
by minimal geodesics of rotation number $r$. For irrational $r \in \mathbb{R}$
all geodesics are minimal, for rational $r \in \mathbb{R} \cup \{\infty\}$ 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.
