A detailed construction of the universal integrability objects related to the integrable systems associated with the quantum group $mathrm U_q(mathcal L(mathfrak{sl}_3))$ is given. The full proof of the functional relations in the form independent of the representation of the quantum group on the quantum space is presented. The case of the general gradation and general twisting is treated. The specialization of the universal functional relations to the case when the quantum space is the state space of a discrete spin chain is described.
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