Solve this system of coupled non-linear equations

I have the following two equations with complex $$\alpha, \beta$$: $$0 = -i \omega_m \beta - ig|\alpha|^2 - \frac{\gamma}{2}\beta$$ $$0 = \alpha(i\Delta - \frac{\kappa}{2}) - i\epsilon - ig\alpha(\beta + \beta^*)$$ The other parameters $$g, \omega_m, \Delta, \kappa, \gamma, \epsilon$$ are real.

To solve this system for $$\alpha$$, I solved the first equation for $$\beta$$ and got $$\beta = \frac{g|\alpha|^2}{\frac{i\gamma}{2} - \omega_m}$$

I then inserted this into the second line and obtained $$\alpha\left(i\Delta - \frac{\kappa}{2} + \frac{2ig^2\omega_m|\alpha|^2}{\frac{\gamma^2}{4} + \omega_m}\right) = i\epsilon$$However, in this equation, both $$\alpha$$ and $$|\alpha|^2$$ appear, and I don't know how to solve this.

The last equation is of the form $$\,\alpha\left(u + i\left(v + w\, |\alpha|^2\right)\right) = i \epsilon\,$$ with $$\,u,v,w,\epsilon \in \mathbb R\,$$.
$$|\alpha|^2\left(u^2 + \left(v+ w\,|\alpha|^2\right)^2\right) = \epsilon^2$$
This is a cubic equation with real coefficients in $$\,|\alpha|^2\,$$ which can be solved either algebraically or numerically. Substituting the positive real solution(s) $$\,|\alpha|^2\,$$ in the first equation of the system gives $$\,\beta\,$$, then substituting the respective $$\,\beta\,$$ in the second equation gives $$\,\alpha\,$$.