  We investigate the existence of Lagrangian fibrations on the generalized
Kummer varieties of Beauville. For a principally polarized abelian surface A of
Picard number one we find the following: The Kummer variety K^n(A) is
birationally equivalent to another irreducible symplectic variety admitting a
Lagrangian fibration, if and only if n is a perfect square. And this is the
case if and only if K^n(A) carries a divisor with vanishing Beauville-Bogomolov
square.
