  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.
