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 metric 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 has constant flag curvature.
"Geodesic behavior for Finsler metrics of constant positive flag curvature on $S^2$." J. Differential Geom. 117 (1) 1 - 22, January 2021. https://doi.org/10.4310/jdg/1609902015