I am currently trying to classify fixed points in my 4D system of ODEs. Null growth isoclines correspond to the following system of algebraic equations $$\alpha_{1}u\left(\frac{L - u - \beta_{1}z - \beta_{2}}{L}\right) + f_{1} = 0$$ $$\alpha_{2}u\left(\frac{M - v - \beta_{3}w - \beta_{4}}{M}\right) + f_{2} = 0$$ $$\alpha_{4}u = \beta_{5}w, w = \frac{\alpha_{4}}{\beta_{5}}u$$ $$\alpha_{6}v = \beta_{6}z, z = \frac{\alpha_{6}}{\beta_{6}}v$$
Here $\alpha_{1}\ldots \beta_{6}, L, M, f_{1}, f_{2} \in R: > 0$ are constant coefficients and $u, v, w, z \in R: > 0$ are the unknowns. Substituting $w$ and $z$ in first two equations yields
$$Au + Bu^2 + Cuv + f_{1} = 0$$ $$Dv + Ev^2 + Fuv + f_{2} = 0$$
Where $A = \alpha_{1} - \frac{\alpha_{1}\beta_{2}}{L}, B = -\frac{\alpha_{1}}{L} < 0, C = -\frac{\alpha_{1}\beta_{1}\alpha_{6}}{L\beta_{6}} < 0, D = \alpha_{2} - \frac{\alpha_{2}\beta_{4}}{M}, E = -\frac{\alpha_{2}}{M} < 0, F = -\frac{\alpha_{2}\beta_{3}\alpha_{4}}{M\beta_{5}} < 0$.
If I can find general solutions for that 2D system of algebraic equations, I can therefore find solutions for $w, z$. However, I cannot find a proper way to solve this. Any help would be greatly appreciated!