On the $\mathfrak{grt}$ hexagon symmetry

In this paper we show that it is possible to project onto the solutions of the $\mathfrak{grt}$ hexagon equation. We also consider in some sense generalized hexagon equations and other symmetry equations for multiple argument maps between groups or torsors as source and show that for these equations we can construct at least some canonical solutions by symmetrization procedures. With help of this solutions we finally introduce a bunch of differentials associated to the considered symmetries.