Rational Integrability of Trigonometric Polynomial Potentials on the Flat Torus

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 rational integrability condition on $V$, using the support of its Fourier transform. We then use this condition to prove that a real trigonometric polynomial potential is rationally integrable if and only if it separates up to rotation of the coordinates. Removing the real condition, we also make a classification of rationally integrable potentials in dimensions $2$ and $3$ and recover several integrable cases. After a complex change of variables, these potentials become real and correspond to generalized Toda integrable potentials. Moreover, along the proof, some of them with high-degree first integrals are explicitly integrated.