= Equal-amplitude states of an eight-mode square pattern
For the four <wavevectors> $(2,1),(2,-1),(1,2),(1,-2)$ and their negatives, use invariant <phases> $\chi_1=2\theta_1-2\theta_2-\theta_3+\theta_4$, $\chi_2=\theta_1+\theta_2-2\theta_3-2\theta_4$. Add to the cubic <normal form of a dynamical system> the two quintic terms $e\overline A_2A_3^2A_4^2$ and $f\overline A_1A_2^2A_3\overline A_4$ in the first equation, together with their square-symmetry partners; $e,f$ are real. For equal positive magnitudes, steadiness forces $\sin\chi_1=\sin\chi_2=0$ unless $e=f=0$. Equality of radial equations further gives $(e-f)(\cos\chi_1-\cos\chi_2)=0$. Consequently the generic <phase> types are $(0,0)$ and $(\pi,\pi)$. When $e=f\ne0$, the mixed type $(0,\pi)$, equivalent by square <symmetry> to $(\pi,0)$, also exists. When both quintic <coefficients> vanish, every <phase> pair is allowed. <Amplitude> existence must still be checked from the remaining scalar equation.
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