Hyper-K\"ahler Fourfolds Fibered by Elliptic Products

Every fibration of a projective hyper-K"ahler fourfold has fibers which are Abelian surfaces. In case the Abelian surface is a Jacobian of a genus two curve, these have been classified by Markushevich. We study those cases where the Abelian surface is a product of two elliptic curves, under some mild genericity hypotheses.