The intermediate jacobian fibration of a cubic fourfold containing a plane and fibrations in Prym varieties

We give a description of the intermediate Jacobian fibration attached to a general complex cubic fourfold X containing a plane as a Lagrangian subfibration of a moduli space of torsion sheaves on the K3 surface associated to X up to a cover. To do so, we propose a general construction of Lagrangian fibrations in Prym varieties as subfibrations of Beauville-Mukai systems over some loci of nodal curves in linear systems on K3 surfaces.