= Solution
Assume $\hat s>0$, as required for the subcritical case. The zero state exists for every $\mu$. For a nonzero uniform real state, put $B=A_0^2>0$. Its <equilibrium point> condition is $\mu+3\hat s B-10B^2=0$, giving
$$
\boxed{B_\pm=\frac{3\hat s\pm\sqrt{9\hat s^2+40\mu}}{20}.}
$$
Both are positive for $-9\hat s^2/40<\mu<0$. At $\mu=0$, the lower branch reaches zero and the upper has $B=3\hat s/10$. For $\mu>0$ only the upper branch is positive. Each positive $B$ represents real states $A_0=\pm\sqrt B$, as well as their constant complex <wave phase> rotations.
The two nonzero branches merge at
$$
\boxed{\mu_{\rm SN}=-\frac{9\hat s^2}{40},\qquad B_{\rm SN}=\frac{3\hat s}{20}.}
$$
The radial <linearization> is $\lambda_a=\mu+9\hat sB-50B^2=2B(3\hat s-20B)$. It is negative on the upper branch and positive on the lower. The zero state is stable for $\mu<0$ and unstable for $\mu>0$. Thus a <subcritical pitchfork bifurcation> branch leaves zero towards negative $\mu$, and its fold connects to the stable finite-amplitude branch. Constant <wave phase> shifts are neutral for nonzero complex states; the stated attraction is in <amplitude>, with spatial <wave phase> <perturbations> damped except for that uniform <symmetry> mode.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/iii/paper-85-amplitude.png]
{title=Uniform cubic-quintic amplitude branches and the maximum real-sideband growth coefficient}
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