HyperK\"ahler Potentials in Cohomogeneity Two

A hyperK\"ahler potential is a function rho that is a K\"ahler potential for each complex structure compatible with the hyperK\"ahler structure. Nilpotent orbits in a complex simple Lie algebra are known to carry hyperK\"ahler metrics admitting such potentials. In this paper, we explicitly calculate the hyperK\"ahler potential when the orbit is of cohomogeneity two. In some cases, we find that this structure lies in a one-parameter family of hyperK\"ahler metrics with K\"ahler potentials, generalising the Eguchi-Hanson metrics in dimension four.