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(i) One-dimensional Bose gases which interact via unscaled integrable pair interactions and are confined in an external potential increasing faster than quadratically undergo a complete generalized Bose-Einstein condensation (BEC) at any temperature, in the sense that a macroscopic number of particles are distributed on a o(N)number of one-particle states. (ii) In a one dimensional harmonic trap the replacement of the oscillator frequency \\omega by \\omega\\ln N/N gives rise to a phase transition at a=\\hbar\\omega\\beta=1 in the noninteracting gas. 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