  We consider two models of interacting Bose gases: a gas of spin one particles
in the ground state of a cubic box and a one-dimension Bose gas with contact
interactions. We show how to calculate exact eigenstates of the corresponding
N-body Hamiltonians. Both models share the property of not leading to the
formation of a Bose-Einstein condensate, even at zero temperature, in the
strict sense of the existence of a single one-particle state with a macroscopic
population. We show that a lot of physical insight can be gained on these two
model systems by using the usual Hartree-Fock mean field approach: in this
approximation, that we test against the exact result, everything happens as if
a single realization of the system was a Bose-Einstein condensate in a state
phi breaking the rotational or translational symmetry, and varying in a random
way for any new experimental realization.
