Abstract:  Given a real lattice $\mathcal{L}$ on $\mathbb{R}^n$, we consider a trigonometric polynomial potential $V$ with frequency vectors in $\mathcal{L}^*$. We find integrability conditions on the convex hull of the support of $\hat{V}$. The study of the interior of the convex hull comes down to the perturbative analysis of $3$ integrable Hamiltonian systems having high degree first integrals. We introduce a notion of complete rational integrability, for which the Liouville tori generically birationally map to $\mathbb{P}^n$ with linear commuting vector fields, solve these systems in this sense, and perform the perturbative analysis to find additional integrability conditions. Then the problem of finding all integrable potentials comes down to a combinatorial problem, i.e. to find all polyhedral tesselations with prescribed edge length of $\mathbb{S}_{n-1}$. We solve this problem in dimension $2$ and $3$, giving a complete list of integrable cases in dimension $2,3$. We conjecture the conditions we found are also sufficient in higher dimension, and that the resulting potentials are moreover completly rationally integrable.