 Cubic equivariants of the full cube symmetry group

Independent coordinate reflections require the $i$th component to be odd in $x_i$ and even in every other coordinate. Permutations force the same two cubic coefficients in all components. Absolute irreducibility forces a scalar linear coefficient, which can be normalized to the transverse bifurcation parameter. No quadratic terms occur. The axial branches with one, two or three equal nonzero absolute coordinates have leading squared amplitude $-\mu /[a+(m-1)b]$, provided that denominator is nonzero.