= Solution
Write $\psi=\theta+\varphi$, $U=\beta_r+\gamma_r$ and $V=\beta_i+\gamma_i$. The polar <amplitude equations> are
$$
\begin{aligned}
\dot R&=R(\mu-\beta_rR^2-\gamma_rS^2)+\nu S\cos\psi,\\
R\dot\theta&=-R(\beta_iR^2+\gamma_iS^2)-\nu S\sin\psi,
\end{aligned}
$$
with their reflected counterparts. On the equal-amplitude <standing wave> branch $R=S>0$, <equilibrium point> requires
$$
Vz+\nu\sin\psi=0,\qquad \mu-Uz+\nu\cos\psi=0,\qquad z=R^2.
$$
Squaring and adding eliminates the phase:
$$
(U^2+V^2)z^2-2\mu Uz+\mu^2-\nu^2=0.
$$
For $U^2+V^2>0$ the candidate squared amplitudes are
$$
\boxed{z_\pm=\frac{\mu U\pm\sqrt{(U^2+V^2)\nu^2-\mu^2V^2}}{U^2+V^2}.}
$$
For each positive root, set $\sin\psi=-Vz/\nu$ and $\cos\psi=(Uz-\mu)/\nu$. The quadratic identity ensures these define a phase. Conversely these phase equations imply the quadratic, so a nonzero phase-locked branch exists exactly when the <discriminant> is nonnegative \b[and at least one root is positive].
Since $|\cos\chi|=|V|/\sqrt{U^2+V^2}$, the <discriminant> condition is the printed $|\mu\cos\chi/\nu|\le1$ for $\nu\ne0$. It is sufficient under the usual above-onset supercritical assumptions $\mu>0,U>0$, because then the plus root is positive. Without these assumptions the condition is not sufficient: $\mu=-1,U=1,V=0,\nu=1/2$ satisfies it but gives $z=-1\pm1/2<0$. This is the <positivity condition for a phase-locked standing wave>, not a removable algebraic detail. The zero solution always exists and is distinct from the nonzero branch. For $\nu=0$, the undivided equations must be used; a nonzero <equilibrium point> in this fixed rotating frame requires $V=0$ and $z=\mu/U>0$. If $U=V=0$, nonzero <equilibrium points> require $\mu^2=\nu^2$ and a compatible phase, with no cubic amplitude selection.
Strong enough forcing can overcome the nonlinear frequency shift, lock the sum phase to the external signal and produce a <standing wave> at half the forcing frequency. The difference phase remains free through <spatial translation>. The amplitude inequality is an existence condition, not a stability theorem: attraction requires the linearized forced <amplitude equations>, and nonlinear coefficients have not been restricted enough here to promise stability of every root.
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