  The high spin ultracold atom models with a special form of contact
interactions, 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, is
studied. It is found that those models have either $Sp(2s+1)$-symmetry for the
fermions or $SO(2s+1)$-symmetry for the bosons in the spin sector. Based on the
symmetry analysis, a new class of exactly solvable models is proposed and
solved via the Bethe ansatz. The ground states for repulsive fermions are also
discussed.
