The authors study an XXZ model of spin-1/2 chains with the quantum sl(2) symmetry, where they reduce the Hamiltonian equation to a set of coupled linear algebraic equations. When q is a root of unity, some eigenstates of the Hamiltonian coincide with each other, and a new state is obtained through a limit process from their difference. This eigenspace corresponds to a reducible but indecomposable representation (type 1) of the quantum sl(2) enveloping algebra.