= Solution
Let $q_j$ denote the four critical <wavevectors>, and write $A_j=R_je^{i\theta_j}$. A spatial translation by $(s,t)$ acts on the <amplitudes> as
$$
(A_1,A_2,A_3,A_4)\longmapsto
(e^{i(2s+t)}A_1,e^{i(2s-t)}A_2,e^{i(s+2t)}A_3,e^{i(s-2t)}A_4).
$$
The two independent translation directions in <phase space> are $(2,2,1,1)$ and $(1,-1,2,-2)$. Their common orthogonal complement is spanned by $(2,-2,-1,1)$ and $(1,1,-2,-2)$. Hence <translation invariants of Fourier-mode phases> can be chosen as
$$
\boxed{\chi_1=2\theta_1-2\theta_2-\theta_3+\theta_4,\qquad
\chi_2=\theta_1+\theta_2-2\theta_3-2\theta_4.}
$$
Away from zero <amplitudes>, a translation-invariant function of the <phases> is a function of these two invariant angles. The four magnitudes are themselves invariant. Thus an <equivariant dynamical system> on the eight-real-dimensional <centre manifold> descends, after quotienting <translation symmetry>, to \b[four magnitude equations and two invariant-phase equations]. The <phase> coordinates are singular where an <amplitude> vanishes; the original complex-amplitude equations remain smooth there.
The <wavevector selection rule for equivariant monomials> proves the cubic regularity. For a <monomial> $\prod_jA_j^{p_j}\overline A_j^{q_j}$ in the first equation, put $d_j=p_j-q_j$. Translation covariance requires
$$
2d_1+2d_2+d_3+d_4=2,\qquad d_1-d_2+2d_3-2d_4=1.
$$
At total degree $N$, additionally $\sum|d_j|\leq N$ and $N-\sum|d_j|$ is even. Conversely, any integer vector satisfying these conditions yields a <monomial>, with any remaining even degree supplied by factors $|A_j|^2$. This makes the degree test exhaustive. For example, eliminate the first two entries:
$$
d_1=\frac{4-5d_3+3d_4}{4},\qquad
d_2=\frac{3d_3-5d_4}{4}.
$$
Checking the integer pairs $|d_3|+|d_4|\leq N$, retaining only integral $d_1,d_2$ with the norm and parity conditions, gives
$$
\begin{array}{c|c}
N&\text{admissible }d\\\hline
1&(1,0,0,0)\\
3&(1,0,0,0)\\
5&(1,0,0,0),\ (-1,2,1,-1),\ (0,-1,2,2).
\end{array}
$$
At degree three the only possibility is therefore $A_1|A_j|^2$, $j=1,\ldots,4$. Square <symmetry> gives the same conclusion for every other equation. All cubic terms are regular.
Two generators of the square <dihedral group> act as follows. <Reflection> in the $x$ axis gives $(A_1,A_2,A_3,A_4)\mapsto(A_2,A_1,A_4,A_3)$, while interchange of $x,y$ gives $(A_3,\overline A_4,A_1,\overline A_2)$. Rotation by $\pi$ conjugates every <amplitude>, forcing the <coefficients> in a steady <normal form of a dynamical system> to be real. With real unfolding parameter $\sigma$ and real <coefficients> $c_1,c_2,c_3,c_4$, define
$$
\begin{aligned}
G_1&=\sigma+c_1|A_1|^2+c_2|A_2|^2+c_3|A_3|^2+c_4|A_4|^2,\\
G_2&=\sigma+c_1|A_2|^2+c_2|A_1|^2+c_3|A_4|^2+c_4|A_3|^2,\\
G_3&=\sigma+c_1|A_3|^2+c_2|A_4|^2+c_3|A_1|^2+c_4|A_2|^2,\\
G_4&=\sigma+c_1|A_4|^2+c_2|A_3|^2+c_3|A_2|^2+c_4|A_1|^2.
\end{aligned}
$$
The \b[most general cubic normal form is $\dot A_j=A_jG_j+O(|A|^5)$]. There are no further <coefficient> identifications: the square-group stabilizer of one of these oblique <wavevectors> is trivial, so its three couplings to the other magnitudes are independent. At cubic order $\dot R_j=R_jG_j$ and all <phases>, including $\chi_1,\chi_2$, are constant.
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