  We prove that the canonical cover of an Enriques surface does not admit
non-trivial Fourier-Mukai partners. We also show that the canonical cover of a
bielliptic surface has at most one non-isomorphic Fourier-Mukai partner. The
first result is then applied to birational Hilbert schemes of points and the
second to birational generalised Kummer varieties. An appendix establishes that
there are no exceptional or spherical objects in the derived category of a
bielliptic surface.
