Problem. To give a concrete example of a geodesically reversible Finsler metric on the two-sphere that is not just the sum of a reversible Finsler metric and an exact $1$-form.

Some background may be useful since the problem is more of a problem in variational calculus than a problem in geometry.

If $M$ is a smooth manifold and $L: TM \rightarrow \mathbb{R}$ is a (Lagrangian) fuction that is smooth outside the zero section and homogeneous of degree one in the velocities (i.e. $L(x,tv) = tL(x,v)$ for every $t > 0$), the variational problem $$ \gamma \mapsto \int_\gamma L $$ is invariant under orientation-preserving reparametrization of the curve $\gamma$. The Lagrangian $L$ is geodesically reversible if changing the orientation of any of its extremals yields another extremal. If the Lagrangian is reversible ($L(x,-v) = L(x,v)$), then it is geodesically reversible, but the converse is not true. For example, an asymmetric norm on $\mathbb{R}^n$ is geodesically reversible, but it is reversible if and only if the norm is symmetric. Another example can be constructed by taking a reversible Lagrangian and adding to it a closed $1$-form considered as a function on the tangent bundle that is linear in the velocities. This last example is somehow "trivial" and I would like to find many examples of geodesically reversible Lagrangians on compact manifolds that are not of this form. On the torus they are easy to construct because we can always compactify the example with the asymmetric norm, but what happens on the sphere?

I can point to a non-solution: If all the geodesics of a geodesically reversible Finsler metric on the $n$-sphere are closed and of the same length, then the metric is the sum of a reversible metric and an exact $1$-form.