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The exact solutions of the $D^{(1)}_3$ model (or the $so(6)$ quantum spinchain) with either periodic or general integrable open boundary conditions areobtained by using the off-diagonal Bethe Ansatz. From the fusion, the completeoperator product identities are obtained, which are sufficient to enable us todetermine spectrum of the system. Eigenvalues of the fused transfer matricesare constructed by the $T-Q$ relations for the periodic case and by theinhomogeneous $T-Q$ one for the non-diagonal boundary reflection case. Thepresent method can be generalized to deal with the $D^{(1)}_{n}$ modeldirectly.

Off-diagonal Bethe Ansatz for the $D^{(1)}_3$ model