= Square-symmetric cubic steady-state normal form
{title2=$\dot A=\mu A-aA^3-bAB^2,\quad\dot B=\mu B-aB^3-bA^2B$}
Independent sign changes and interchange of two coordinates in the natural <dihedral group> action force the displayed cubic <normal form>. Apart from the origin, its nondegenerate small <equilibria> lie on coordinate axes or on the diagonals. An axial branch has squared amplitude $\mu/a$ and <eigenvalues> $-2\mu,\mu(1-b/a)$. A diagonal branch has component squared amplitude $\mu/(a+b)$ and <eigenvalues> $-2\mu,2\mu(b-a)/(a+b)$. Subtracting the two nonzero equilibrium equations gives $(a-b)(A^2-B^2)=0$, excluding other cubic branches when $a\ne b$.
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