  We study non-reversible Finsler metrics with constant flag curvature 1 on S^2
and show that the geodesic flow of every such metric is conjugate to that of
one of Katok's examples, which form a 1-parameter family. In particular, the
length of the shortest closed geodesic is a complete invariant of the geodesic
flow. We also show, in any dimension, that the geodesic flow of a Finsler
metrics with constant positive flag curvature is completely integrable.
  Finally, we give an example of a Finsler metric on~$S^2$ with positive flag
curvature such that no two closed geodesics intersect and show that this is not
possible when the metric is reversible or have constant flag curvature
