  For an abelian surface $A$, we explicitly construct two new families of
stable vector bundles on the generalized Kummer variety $K_n(A)$ for
$n\geqslant 2$. The first is the family of tautological bundles associated to
stable bundles on $A$, and the second is the family of the "wrong-way" fibers
of a universal family of stable bundles on the dual abelian surface
$\widehat{A}$ parametrized by $K_n(A)$. Each family exhibits a smooth connected
component in the moduli space of stable bundles on $K_n(A)$, which is
holomorphic symplectic but not simply connected, contrary to the case of K3
surfaces.
