The matrix balls consisting of -matrices of norm over are considered. These balls are one possible realization of the symmetric spaces . Generalized linear-fractional maps are maps of the form (they are in general neither injective nor surjective). Characterizations of generalized linear-fractional maps in the spirit of the “fundamental theorem of projective geometry” are obtained: for a certain family of submanifolds of (“quasilines”) it is shown that maps taking quasilines to quasilines are generalized linear-fractional. In addition, for the standard field of cones on (described by the inequality ) it is shown that maps taking cones to cones are generalized linear-fractional.