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Can a Bose-Einstein condensate be made of any boson, or even two different types of bosons. Since none have to obey the Pauli exclusion principle, if so, how can we predict how one would like?

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even two different types of bosons

No.

While any particle that has integer spin is indeed a boson, you need two identical bosons to display the peculiar and characteristic quantum statistical effects.

In truest sense of the word, identical means that they transform in the same representation of the Poincare' group, uniquely labelled by mass $m$ and spin $s$ (by extension, angular momentum).

So a BEC can be formed of particles with the same mass and the same internal state. You can of course have a mixture of particles where each "subset" forms its own BEC though.

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