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We construct a Darboux transformation of a general $su(3)$-valued spin systemcalled the $\Gamma$-spin system. Using this Darboux transformation we derive arecursive formula for the soliton solutions of this spin system. Then usingthese results we present explicit expressions for the 1-soliton solution of thecoupled $su(2)$-valued spin systems, namely, of the coupled M-LIII equation.

Darboux transformations and exact soliton solutions of integrable coupled spin systems related with the Manakov system