  A general method is proposed for constructing the Bethe ansatz equations of
integrable models without U(1) symmetry. As an example, the exact spectrum of
the XXZ spin ring with M{\" o}bius like topological boundary condition is
derived by constructing a modified T-Q relation based on the functional
connection between the eigenvalues of the transfer matrix and the quantum
determinant of the monodromy matrix. With the exact solution, the elementary
excitations of the topological XX spin ring is discussed in detail. It is found
that the excitation spectrum indeed shows a nontrivial topological nature.
