Einstein metrics on bundles over hyperK\"ahler manifolds

We construct explicit examples of quaternion-K"ahler and hypercomplex structures on bundles over hyperK"ahler manifolds. We study the infinitesimal symmetries of these examples and the associated Galicki-Lawson quaternion-K"ahler moment map. By performing the QK reduction we produce several explicit QK metrics. Moreover we are led to a new proof of a hyperK"ahler/quaternion-K"ahler type correspondence. We also give examples of other Einstein metrics and balanced Hermitian structures on these bundles.