Rational $Q$-systems for integrable spin chains without $U(1)$ symmetry

The Q Q -system is an efficient method for finding complete physical solutions of Bethe Ansatz equations, but so far its application has been confined to systems possessing U(1) U ( 1 ) symmetry. We extend the rational Q Q -system framework to integrable spin chains without U(1) U ( 1 ) symmetry, exemplified by the closed XXZ model with anti-diagonal twists and the open XXZ model with non-diagonal boundary fields. We demonstrate that the Q Q -system can be derived by combining TQ T Q -relation with fusion relations of higher-spin transfer matrices. This yields QQ Q Q -relations analogous to the U(1) U ( 1 ) symmetric case but incorporating additional inhomogeneous terms. We present numerical solutions that are validated against exact diagonalization, confirming that it generates all and exclusively physical solutions.