HyperKähler and quaternionic Kähler manifolds with S1 -symmetries

We study relations between quaternionic Riemannian manifolds admitting different types of symmetries. We show that any hyperKähler manifold admitting hyperKähler potential and triholomorphic action of S1S1 can be constructed from another hyperKähler manifold (of lower dimension) with an action of S1S1 that fixes one complex structure and rotates the other two and vice versa. We also study the corresponding quaternionic Kähler manifolds equipped with a quaternionic Kähler action of the circle. In particular we show that any positive quaternionic Kähler manifolds with S1S1-symmetry admits a Kähler metric on an open everywhere dense subset.