  Let O be a nilpotent orbit in g^C where G is a compact, simple group and
g=Lie(G). It is known that O carries a unique G-invariant hyperK\"ahler metric
admitting a hyperK\"ahler potential compatible with the
Kirillov-Kostant-Souriau symplectic form. In this work, the hyperK\"ahler
potential is explicitly calculated when O is of cohomogeneity three under the
action of G. It is found that such a structure lies on a one-parameter family
of hyperK\"ahler metrics with G-invariant K\"ahler potentials if and only if g
is sp(3), su(6), so(7), so(12) or e_7 and otherwise is the unique G-invariant
hyperK\"ahler metric with G-invariant K\"ahler potential.
