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