  We investigate the behaviour of hard-core bosons in one- and two-dimensional
flat band systems, the chequerboard and the kagom\'e lattice and
one-dimensional analogues thereof. The one dimensional systems have an exact
local reflection symmetry which allows for exact results. We show that above
the critical density an additional particle forms a pair with one of the other
bosons and that the pair is localised. In the two-dimensional systems exact
results are not available but variational results indicate a similar physical
behaviour.
