The degree-two bound for irreducible polynomial integrals on the two-torus

Let the geodesic flow be the Hamiltonian flow of a Riemannian metric on the 2-torus. A polynomial first integral is a polynomial in the momenta that Poisson-commutes with the Hamiltonian, and it is irreducible if it is not generated as a nontrivial polynomial expression by lower-degree first integrals. The degree-two bound conjecture. The largest possible degree of any irreducible polynomial first integral of the geodesic flow on the 2-torus is not larger than $2$. This conjecture concerns the global classification of integrable geodesic flows on compact surfaces; the source presents it as a known conjecture and does not provide evidence of a resolution.