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