We explore the S-matrices of gapped, unitary, Lorentz invariant quantum field theories with a global O(N) symmetry in 1+1 dimensions. We extremize various couplings in the two-to-two scattering amplitudes of vector particles. Saturating these bounds, we encounter known integrable models. For general mass spectra -away from the integrable realm- we find (numerically, through a large N analysis and sometimes even analytically) that the S-matrices saturating the various coupling bounds have an extremely rich structure exhibiting infinite resonances and virtual states in the various kinematical sheets. \\ \\ Based on 1805.11143.