  We study symplectic resolutions of the Hilbert scheme of two points on a
surface with one ADE-singularity. We also characterize such singularities by
central fibers of their symplectic resolutions. As an application, we show that
these singularities are isomorphic to the Slodowy slices which are transversal
to the `sub-subregular' orbits in the nilpotent cones of ADE-types.
