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We find the all solutions to the $sl_q(2)$-invariant multi-parametricYang-Baxter equations (YBE) at $q=i$ defined on the cyclic (semi-cyclic,nilpotent) representations of the algebra. We are deriving the solutions inform of the linear combinations over the $sl_q(2)$-invariant objects -projectors. The direct construction of the projector operators at roots ofunity gives us an opportunity to consider all the possible cases, includingalso degenerated one, when the number of the projectors becomes larger, andvarious type of solutions are arising, and as well as the inhomogeneous case.We are giving a full classification of the YBE solutions for the consideredrepresentations. A specific character of the solutions is the existence of thearbitrary functions.

New solutions to the $s\ell_q(2)$-invariant Yang-Baxter equations at roots of unity: cyclic representations