2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/56682We introduce two algebraic completely integrable analogues of the Mumford systems which we call hyperelliptic Prym systems, because every hyperelliptic Prym variety appears as a fiber of their momentum map. As an application we show that the generic fiber of the momentum map of the periodic Volterra lattice $$\dot a_i=a_i(a_{i-1}-a_{i+1}), \qquad i=1,...,n,\quad a_{n+1}=a_1,$$ is an affine part of a hyperelliptic Prym variety, obtained by removing $n$ translates of the theta divisor, and we conclude that this integrable system is algebraic completely integrable.Final version. To appear in CMPMathematical PhysicsAlgebraic GeometryExactly Solvable and Integrable Systems35Q58, 37J35, 58J72, 70H06text