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The high spin ultracold atom models with a special form of contactinteractions, i.e., the scattering lengthes in the total spin-$2,4 \cdots$channels are equal but may be different from that in the spin-0 channel, isstudied. It is found that those models have either $Sp(2s+1)$-symmetry for thefermions or $SO(2s+1)$-symmetry for the bosons in the spin sector. Based on thesymmetry analysis, a new class of exactly solvable models is proposed andsolved via the Bethe ansatz. The ground states for repulsive fermions are alsodiscussed.

Hidden Sp(2s+1)- or SO(2s+1)-symmetry and new exactly solvable models in ultracold atomic systems