Rational integrability of trigonometric polynomial potentials on the flat torus

Abstract

We consider a lattice $\mathcal{L}\subset {\mathbb{R}}^n$ and a trigonometric potential $V$ with frequencies $k\in \mathcal{L}$. We then prove a strong integrability condition on $V$, using the support of its Fourrier transform. We then use this condition to prove that a real trigonometric polynomial potential is integrable if and only if it separates up to rotation of the coordinates. Removing the real condition, we also make a classification of integrable potentials in dimension $2$ and $3$, and recover several integrable cases. These potentials after a complex variable change become real, and correspond to generalized Toda integrable potentials. Moreover, along the proof, some of them with high degree first integrals are explicitly integrated.

29 pages

References in corpus (1)

• The last integrable case of kozlov-Treshchev Birkhoff integrable potentials

Cited by in corpus (3)

• Integrable magnetic geodesic flows on 2-surfaces
• Deformational rigidity of integrable metrics on the torus
• Rational integrals of 2-dimensional geodesic flows: new examples