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.

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.

Added on 21/06/2021.

1. The only examples of geodesically reversible $C^2$ Finsler metrics on closed manifolds that I know are still sums of flat metrics and closed $1$-forms or sums of reversible metrics and closed $1$-forms.

2. Let $M$ be a two- or three-dimensional manifold and let $F$ be a $C^2$ geodesically reversible Finsler metric on M. If the geodesic flow does not admit any non-trivial continuous integral of motion, then the metric $F$ is necessarily the sum of a reversible metric and a closed 1-form.

If the Nakajima-Schneider conjecture on bodies of constant breadth and constant brightness in relative geometry is true in dimensions greater than three, then the preceding result extends to all dimensions.

3. If $M$ is a closed surface of genus greater than one and $F$ is a real-analytic, geodesically-reversible Finsler metric, then $F$ is the sum of a reversible metric and a closed $1$-form.

4. Let $F$ be a $C^2$ geodesically-reversible Finsler metric on a manifold $M$. If every geodesic passes through some open set where $F$ is reversible, then $F$ is the sum of a reversible metric and a closed $1$-form.

See Rigidity results for geodesically reversible Finsler metrics