Cubic equivariants of the full cube symmetry group (${\dot{x}}_i=\mu x_i+a{x}_i^3+b{x}_i{\sum }_{j\ne i}{x}_j^2+\cdots$)

For the natural action of all signed coordinate permutations on ${\mathbb{R}}^3$, a cubic equivariant dynamical system has componentsReflecting $x_i$ alone reverses component $i$, so that component must be an odd function of $x_i$. Reflecting any other coordinate leaves component $i$ fixed, so it must be an even function of each other coordinate. Consequently the only cubic monomials are $x_i^3$ and $x_i{x}_j^2$. Permuting the other coordinates makes their coefficients equal; permuting all coordinates gives the same pair $a,b$ in every component. The same parity test excludes every quadratic term.

A branch with exactly $m\in \{1,2,3\}$ nonzero coordinates of common magnitude $r$ satisfiesThus its nonzero states require $r^2=-\mu /[a+(m-1)b]>0$, provided the denominator is nonzero. The three cases describe coordinate-axis, face-diagonal and body-diagonal states. If the denominator vanishes, the cubic truncation cannot determine that branch's amplitude; higher-order terms must be retained. Existence of a branch does not by itself determine its stability.