  The Hubbard model of bosons on two dimensional lattices with a lowest flat
band is discussed. In these systems there is a critical density, where the
ground state is known exactly and can be represented as a charge density wave.
Above this critical filling, depending on the lattice structure and the
interaction strength, the additional particles are either delocalised and
condensate in the ground state, or they form pairs. Pairs occur at strong
interactions, e.g., on the chequerboard lattice. The general mechanism behind
this phenomenon is discussed.
