2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/159488We study Beauville's completely integrable system and its variant from a viewpoint of multi-Hamiltonian structures. We also relate our result to the previously known Poisson structures on the Mumford system and the even Mumford system.This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/Mathematical PhysicsExactly Solvable and Integrable Systems37J35; 14H70; 14H40text