  We study the exact solution of quantum integrable system associated with the
$A^{(2)}_3$ twist Lie algebra, where the boundary reflection matrices have
non-diagonal elements thus the $U(1)$ symmetry is broken. With the help of the
fusion technique, we obtain the closed recursive relations of the fused
transfer matrices. Based on them, together with the asymptotic behaviors and
the values at special points, we obtain the eigenvalues and Bethe ansatz
equations of the system. We also show that the method is universal and valid
for the periodic boundary condition where the $U(1)$ symmetry is reserved. The
results in this paper can be applied to studying the exact solution of the
$A^{(2)}_n$-related integrable models with arbitrary $n$.
