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Let M be a closed orientable surface of negative curvature. A connection is said to be transparent if its parallel transport along closed geodesics is the identity. We describe all transparent SU (2)-connections and we show that they can be built up from suitable Bäcklund transformations.doi:10.1515/crelle.2011.058 fatcat:hqznsoikivfrpb2iun7p7eom34