  We study one-dimensional truncated (no more than 2 particles on a site)
bosonic Hubbard model in both repulsive and attractive regimes by exact
diagonalization and exact worldline Monte Carlo simulation. In the commensurate
case (one particle per site) we demonstrate that the point of Mott-insulator --
superfluid transition, $(U/t)_c=0.50\pm 0.05$, is remarkably far from that of
the full model. In the attractive region we observe the phase transition from
one-particle superfluid to two-particle one. The paring gap demonstrates a
linear behavior in the vicinity of the critical point. The critical state
features marginal response to the gauge phase. We argue that the two-particle
superfluid is a macroscopic analog of a peculiar phase observed earlier in a
spin-1 model with axial anisotropy.
