Lagrangian fibrations from Fano varieties of cubic fourfolds

We show that for a general cubic fourfold $X$ containing a plane $P$, the Mukai flop of its Fano variety of lines $F$ along the dual plane $P^*$ admits a map to $\mathbb{P}^2$ fibered in geometrically abelian surfaces. The explicit construction of this fibration affords an analysis of the torsor structure of the smooth fibers when the ground field $k$ is not closed