  We prove that the Fano variety of lines of a generic cubic fourfold
containing a plane is isomorphic to a moduli space of twisted stable complexes
on a K3 surface. On the other hand, we show that the Fano varieties are always
birational to moduli spaces of twisted stable coherent sheaves on a K3 surface.
The moduli spaces of complexes and of sheaves are related by wall-crossing in
the derived category of twisted sheaves on the corresponding K3 surface.
