  We study the exact solution of an $N$-state vertex model based on the
representation of the $U_q[SU(2)]$ algebra at roots of unity with diagonal open
boundaries. We find that the respective reflection equation provides us one
general class of diagonal $K$-matrices having one free-parameter. We determine
the eigenvalues of the double-row transfer matrix and the respective Bethe
ansatz equation within the algebraic Bethe ansatz framework. The structure of
the Bethe ansatz equation combine a pseudomomenta function depending on a
free-parameter with scattering phase-shifts that are fixed by the roots of
unity and boundary variables.
