  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.
