We perform a systematic exact algebraic search for integrable spin-S chains which are isotropic in spin space, i.e. are su(2)-invariant. The families of spin chains found for S ≤ 13.5 support recent arguments in favour of the complete classification of all such integrable chains. The integrable families of spin chains are discussed in the light of the conjectured spin-dependent properties of the Heisenberg chain.
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