  The S-matrices for the scattering of two excitations in the XYZ model and in
all of its SU(n)-type generalizations are obtained from the asymptotic behavior
of Kerov's generalized Hall-Littlewood polynomials. These physical scattering
processes are all reduced to geometric s-wave scattering problems on certain
quantum-symmetric spaces, whose zonal spherical functions these
Hall-Littlewood-Kerov polynomials are. Mathematically, this involves a
generalization with an unlimited number of parameters of the Macdonald
polynomials. Physically, our results suggest that, of the (1+1)-dimensional
models, the integrable ones are those, for which the scattering of excitations
becomes geometric in the sense above.
