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We present a simple procedure to construct non-local conserved charges forclassical open strings on coset spaces. This is done by including suitablereflection matrices on the classical transfer matrix. The reflection matricesmust obey certain conditions for the charges to be conserved and in involution.We then study bosonic open strings on $AdS_5\times S^5$. We consider boundaryconditions corresponding to Giant Gravitons on $S^5$, $AdS_4\times S^2$D5-branes and $AdS_5 \times S^3$ D7-branes. We find that we can construct theconserved charges for the full bosonic string on a Maximal Giant Graviton or aD7-brane. For the D5-brane, we find that this is possible only in a SU(2)sub-sector of the open string. Moreover, the charges can not be constructed atall for non-maximal Giant Gravitons. We discuss the interpretation of theseresults in terms of the dual gauge theory spin chains.Comment: 23 pages, JHEP styl

Classical open string integrability