The integrable (hyper)eclectic spin chain

We refine the recently introduced notion of eclectic spin chains by including a maximal number deformation parameters. These models are integrable, nearest-neighbor n-state with exceedingly simple non-hermitian Hamiltonians. They turn out to be non-diagonalizable in multiparticle sector (n>2), where their "spectrum" consists an intricate collection Jordan blocks arbitrary size and multiplicity. show how why quantum inverse scattering method, sought universally applicable integrable chains, essentially fails reproduce details this spectrum. then provide, for n=3, detailed evidence variety analytical numerical techniques that spectrum is not "random", but instead shows surprisingly subtle regular patterns moreover exhibit universality generic also introduce new model, hypereclectic chain, all parameters zero except one. Despite extreme simplicity its Hamiltonian, it still seems above "generic" spectra as subset even more overall Our inspired parts one-loop dilatation operator strongly twisted, double-scaled N=4 Super Yang-Mills Theory.