  We show that the regular Slodowy slice to the sum of two semisimple adjoint
orbits of $GL(n,C)$ is isomorphic to the deformation of the $D_2$-singularity
if $n=2$, the Dancer deformation of the double cover of the Atiyah-Hitchin
manifold if $n=3$, and to the Atiyah-Hitchin manifold itself if $n=4$. For
higher $n$, such slices to the sum of two orbits, each having only two distinct
eigenvalues, are either empty or biholomorphic to open subsets of the Hilbert
scheme of points on of one the above surfaces. In particular, these open
subsets of Hilbert schemes of points carry complete hyperk\"ahler metrics. In
the case of the double cover of the Atiyah-Hitchin manifold this turns out to
be the natural $L^2$-metric on a hyperk\"ahler submanifold of the monopole
moduli space.
