TODA FIELDS OF SO(3) HYPER-KAHLER METRICS AND FREE-FIELD REALIZATIONS

The Eguchi-Hanson, Taub-NUT and Atiyah-Hitchin metrics are the only co mplete nonsingular SO(3)-invariant hyper-Kahler metrics in four dimens ions. The presence of a rotational SO(2) isometry allows their unified treatment based on solutions of the 3D continual Toda equation. We de termine the Toda potential in each case and examine the free held real ization of the corresponding solutions, using infinite power series ex pansions. The Atiyah-Hitchin metric exhibits some unusual features att ributed to topological properties of the group of area preserving diff eomorphisms. The construction of a descending series of SO(a)-invarian t 4D regular hyper-Kahler metrics remains an interesting question.