The Frenkel-Kontorova model is a standard model in condensed matter physics describing particles having nearest-neighbor spring-like interactions. Mathematical analysis of this model leads to studying standard-like twist maps. In the 80s, Suris found a remarkable family of potentials for this model with integrable dynamics. In some sense, this is similar to the role that ellipses play in planar billiards. In the talk, we will highlight this connection via the action-angle coordinates of the two systems. Then we will also show that an integrable perturbation of a Suris potential has to be a Suris potential itself. This is in the spirit of local results proven for the Birkhoff conjecture in billiards. The proof relies heavily on Fourier analysis, as well as the construction of a suitable basis for L2, which captures the dynamics of the system. Joint work with Corentin Fierobe. Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023–2027).