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We construct an integrable quantum spin chain that includes the nearest-neighbor,next-nearest-neighbor,chiral three-spin couplings,Dzyloshinsky-Moriya interactions and unparallel boundary magnetic fields.Although the interactions in bulk materials are isotropic,the spins nearby the boundary fields are polarized,which induce the anisotropic exchanging interactions of the first and last bonds.The U (1) symmetry of the system is broken because of the off-diagonal boundary reflections.Using the off-diagonal Bethe ansatz,we obtain an exact solution to the system.The inhomogeneous T-Q relation and Bethe ansatz equations are given explicitly.We also calculate the ground state energy.The method given in this paper provides a general way to construct new integrable models with certain interesting interactions.