= Solution
Take the leading real field to be $A e^{i(k_cx+\omega_0t)}+B e^{i(-k_cx+\omega_0t)}+\text{complex conjugate}$. A <spatial translation> gives $(A,B)\mapsto(e^{ik_c\delta}A,e^{-ik_c\delta}B)$; a time translation gives $(A,B)\mapsto(e^{i\psi}A,e^{i\psi}B)$; spatial reflection exchanges $A$ and $B$.
For a monomial $A^j\overline A^kB^l\overline B^m$ in the first <amplitude equation>, covariance requires $j-k+l-m=1$ and $j-k-l+m=1$. Hence $j-k=1$, $l-m=0$. At linear order only $A$ is allowed, no quadratic monomial satisfies these conditions, and the cubic possibilities are $A^2\overline A$ and $AB\overline B$. Reflection fixes the coefficients in the second equation. Removing the small common linear frequency correction by a rotating frame makes $\mu$ real. This derives the two cubic <amplitude equations>, rather than merely postulating them.
Let $r=|A|^2$, $s=|B|^2$. Taking real parts gives
$$
\dot r=2r(\mu-\beta_r r-\gamma_r s),\qquad
\dot s=2s(\mu-\beta_r s-\gamma_r r).
$$
A <travelling wave> has $(r,s)=(\mu/\beta_r,0)$, or its reflection, whenever $\mu/\beta_r>0$. Its phase evolves at $-\beta_i\mu/\beta_r$, shifting the physical oscillation frequency. The nonzero-amplitude radial <eigenvalue> is $-2\mu$, while the absent mode has real growth rate $\mu(1-\gamma_r/\beta_r)$. Therefore, on the supercritical side,
$$
\boxed{\text{stable travelling wave: }\mu>0,\quad\beta_r>0,\quad\gamma_r>\beta_r.}
$$
The phase direction is neutral because of the symmetry, so stability means <orbital stability>. A branch with $\mu<0$ is radially unstable, whatever the transverse coefficient.
A <standing wave> has $r=s=h=\mu/(\beta_r+\gamma_r)>0$. Its two phases evolve together at $-(\beta_i+\gamma_i)h$; the relative phase fixes the spatial position of the standing pattern. In the two intensity directions the <eigenvalues> are $-2\mu$ and $-2h(\beta_r-\gamma_r)$. Thus
$$
\boxed{\text{stable standing wave: }\mu>0,\quad\beta_r+\gamma_r>0,\quad\beta_r>\gamma_r.}
$$
Again the symmetry phases are neutral and the conclusion is <orbital stability>. Equality $\beta_r=\gamma_r$ leaves a continuum of mixed intensities at cubic order; equality $\beta_r+\gamma_r=0$ removes cubic standing-wave saturation. These degeneracies require higher-order terms and cannot be classified by the cubic truncation. The zero state is linearly stable for $\mu<0$ and unstable for $\mu>0$.
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