  We present a phase diagram for a dilute two-dimensional Bose-gas on a
lattice. For one sort of boson we consider a realistic case of the van der
Waals interaction between particles with a strong hard-core repulsion $U$ and a
van der Waals attractive tail $V$. For $V< 2 t $, $t$ being a hopping
amplitude, the phase diagram of the system contains regions of the usual
one-particle Bose-Einstein condensation (BEC). However for $V>2t$ we have total
phase separation on a Mott-Hubbard Bose solid and a dilute Bose gas. For two
sorts of structureless bosons described by the two band Hubbard model an s-wave
pairing of the two bosons of different sort $<b_1 b_2 > \neq 0$ is possible.
The results we obtained should be important for different Bose systems,
including submonolayers of $^4$He, excitons in semiconductors, Schwinger bosons
in magnetic systems and holons in HTSC. In the HTSC case a possibility of
two-holon pairing in the slave-bosons theories of superconductivity can restore
a required charge $2e$ of a Cooper pair.
