= Solution
For a spatially uniform pump and $\beta>0$, a nonzero homogeneous solution has constant <number density> determined by gain saturation. Its <complex argument> rotates uniformly:
$$
\boxed{\Psi_\infty(t)=\sqrt{n_\infty}\,e^{-i\mu t+i\theta_0},\qquad
n_\infty=\frac\alpha\beta,\qquad \mu=gn_\infty-s.}
$$
This <uniform pumped polariton condensate> requires $\alpha>0$, equivalently
$$
P>P_{\rm th}=\frac{\gamma_C\gamma_R}{R_R}.
$$
In the cubic approximation its <number density> is
$$
n_\infty=\frac{\gamma_R}{R_R}\left(1-\frac{P_{\rm th}}P\right).
$$
It is not the exact <number density> of the original two-field equations, which would be $(P-P_{\rm th})/\gamma_C$: the two coincide to first order in distance above threshold. At or below threshold there is no positive nonzero branch of this form; the vacuum remains a solution. This algebra establishes existence, not stability for every possible sign of $g$.
Choose $\theta_0=0$ and seek a time-independent normalized profile through
$$
\Psi(\mathbf r,t)=\sqrt{n_\infty}\,e^{-i\mu t}\psi(\mathbf r).
$$
Using $\beta n_\infty=\alpha$ and $\mu+s=gn_\infty$, its equation becomes
$$
0=\alpha(1-|\psi|^2)\psi+i\left[\nabla^2+gn_\infty(1-|\psi|^2)\right]\psi.
$$
Division by $i$ therefore yields
$$
\boxed{\left[\nabla^2+\xi(1-|\psi|^2)\right]\psi=0,\qquad
\xi=(g-i\beta)n_\infty=gn_\infty-i\alpha.}
$$
For a localized zero-winding disturbance the bulk condition is $\psi\to1$. Here $\xi$ is a complex coefficient with inverse-length-squared dimensions in the scaled equation, not the real wall-healing length denoted by the same letter in Question 1. The <constant far-field phase excludes net vortex winding> lemma explains why the next part requires interpreting the vortex boundary differently.
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