2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/60379By studying completely integrable torus actions on contact manifolds we prove a conjecture of Toth and Zelditch that toric integrable geodesic flows on tori must have flat metrics.12 pages. V2: paper shortened to 7 pages.V3: final version; introduction expanded; to appear in Math Res. LettersDifferential GeometryAlgebraic GeometryDynamical Systemstext