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We consider an open spin chain model with GL(N) bulk symmetry that is brokento GL(M) x GL(N-M) by the boundary, which is a generalization of a modelarising in string/gauge theory. We prove the integrability of this model byconstructing the corresponding commuting transfer matrix. This constructionuses operator-valued "projected" K-matrices. We solve this model for generalvalues of N and M using the nested algebraic Bethe ansatz approach, despite thefact that the K-matrices are not diagonal. The key to obtaining this solutionis an identity based on a certain factorization property of the reducedK-matrices into products of R-matrices. Numerical evidence suggests that thesolution is complete.

Nested algebraic Bethe ansatz for open GL(N) spin chains with projected K-matrices