Toda fields of SO(3) hyper-Kahler metrics and free field realizations

The Eguchi-Hanson, Taub-NUT and Atiyah-Hitchin metrics are the only complete nonsingular SO(3)-invariant hyper-Kahler metrics in four dimensions. The presence of a rotational SO(2) isometry allows their unified treatment based on solutions of the 3D continual Toda equation. We determine the Toda potential in each case and examine the free held realization of the corresponding solutions, using infinite power series expansions. The Atiyah-Hitchin metric exhibits some unusual features attributed to topological properties of the group of area preserving diffeomorphisms. The construction of a descending series of SO(a)-invariant 4D regular hyper-Kahler metrics remains an interesting question.