# Rational integrability of trigonometric polynomial potentials on the flat torus

## TL;DR

This paper establishes a strong integrability criterion for trigonometric polynomial potentials on flat tori, classifies integrable cases in low dimensions, and connects them to generalized Toda systems.

## Contribution

It introduces a Fourier support-based integrability condition and classifies integrable potentials in dimensions 2 and 3, linking them to Toda-like systems.

## Key findings

- Integrable potentials are characterized by their Fourier support.
- Real integrable potentials are those that are separable up to rotation.
- Explicit integration of some high-degree first integrals was achieved.

## 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.