Maupertuis' conjecture for cubic polynomial integrals on the torus

Let $T^2$ be the two-dimensional torus with a Riemannian metric, and consider its geodesic flow. A polynomial integral of degree $3$ in momenta is a nontrivial function on the cotangent bundle that is preserved by this flow. Maupertuis' conjecture. If the geodesic flow admits a nontrivial integral polynomial of degree $3$ in momenta, then the metric admits a Killing vector field. The conjecture is presented as a special case of the problem of classifying polynomial integrals on closed surfaces; its resolution status is not specified in the source.