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Initially, we derive a nonlinear integral equation for the vacuum countingfunction of the spin 1/2-XYZ chain in the {\it disordered regime}, thusparalleling similar results by Kl\"umper \cite{KLU}, achieved through adifferent technique in the {\it antiferroelectric regime}. In terms of thecounting function we obtain the usual physical quantities, like the energy andthe transfer matrix (eigenvalues). Then, we introduce a double scaling limitwhich appears to describe the sine-Gordon theory on cylindrical geometry, sogeneralising famous results in the plane by Luther \cite{LUT} and Johnson etal. \cite{JKM}. Furthermore, after extending the nonlinear integral equation toexcitations, we derive scattering amplitudes involving solitons/antisolitonsfirst, and bound states later. The latter case comes out as manifestly relatedto the Deformed Virasoro Algebra of Shiraishi et al. \cite{SKAO}. Although thisnonlinear integral equations framework was contrived to deal with finitegeometries, we prove it to be effective for discovering or rediscoveringS-matrices. As a particular example, we prove that this unique model furnishesexplicitly two S-matrices, proposed respectively by Zamolodchikov \cite{ZAMe}and Lukyanov-Mussardo-Penati \cite{LUK, MP} as plausible scattering descriptionof unknown integrable field theories.Comment: Article, 41 pages, Late

From finite geometry exact quantities to (elliptic) scattering amplitudes for spin chains: the 1/2-XYZ