Exactly soluble model with two kinds of Bose–Einstein condensations

We present a simple exactly soluble model which manifests two kinds of Bose–Einstein condensations. They occur in two stages: for intermediate densities (negative chemical potentials μ0<μ<0) one has a non-conventional Bose condensation in the mode k=0 due to a diagonal perturbation of the perfect Bose-gas (PBG), whereas for μ=0 and large densities ρ>ρc0(θ) one gets a conventional (generalized) Bose-condensation corresponding to the standard mechanism of saturation.