Regular and Chaotic Dynamics

, Volume 22, Issue 4, pp 386–407 | Cite as

Rational integrability of trigonometric polynomial potentials on the flat torus

  • Thierry CombotEmail author


We consider a lattice ℒ ⊂ ℝ n and a trigonometric potential V with frequencies k ∈ 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.


trigonometric polynomials differential Galois theory integrability Toda lattice 

MSC2010 numbers



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© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Scuola Normale SuperioreCentro di Ricerca Matematica Ennio De Giorgi, Laboratorio FibonacciPiazza CavalieriPisaItaly

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