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Coordinate reflection in $x_i$ requires the $i$th component to be odd in $x_i$, and reflections in the other coordinates require it to be even in those coordinates. Thus no constant or quadratic terms are permitted. The only cubic monomials in component $i$ are $x_i^3$ and $x_i{x}_j^2$ for $j\ne i$. Coordinate permutations equate their coefficients in all components. After using the transverse linear eigenvalue as parameter, the normal form isThe remainder starts at fifth spatial order, together with parameter-dependent corrections such as $\mu |x{|}^3$. This is the cubic equivariants of the full cube symmetry group form. On a branch with $m=1,2,3$ equal nonzero absolute coordinates $s$, respectively,Hence the three leading branch amplitudes are $s^2=-\mu /a$, $-\mu /(a+b)$ and $-\mu /(a+2b)$. Their generic nonzero denominators determine on which side the branches occur; existence alone does not make them stable.