  The transfer matrix of the general integrable open XXZ quantum spin chain
obeys certain functional relations at roots of unity. By exploiting these
functional relations, we determine the Bethe Ansatz solution for the transfer
matrix eigenvalues for the special cases that all but one of the boundary
parameters are zero, and the bulk anisotropy parameter is \eta = i\pi/3, i\pi/5
,... In an Addendum, these results are extended to the cases that any two of
the boundary parameters {\alpha_-, \alpha_+,\beta_-, \beta_+} are arbitrary and
the remaining boundary parameters are either \eta or i \pi/2.
