Chiral coordinate Bethe ansatz for phantom eigenstates in the open XXZ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>spin</mml:mi><mml:mtext>−</mml:mtext><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:math> chain

We construct the coordinate Bethe ansatz for all eigenstates of open $\mathrm{spin}\text{\ensuremath{-}}\frac{1}{2}$ XXZ chain that fulfill phantom roots criterion (PRC). Under PRC, Hilbert space splits into two invariant subspaces and there are sets homogeneous equations (BAE) to characterize in each case. propose vectors with chiral shocks span expand corresponding eigenstates. All factorized have symmetrical simple structures. Using several cases as examples, we present core elements our generalized method. The expanded generating set show clear chirality certain symmetry properties. bulk scattering matrices, reflection matrices on boundaries BAE obtained, which demonstrates agreement other approaches. Some hypotheses formulated generalization approach.