Exploring Steady States for DEs: $(0,0)$ and $(1,0)$?

[Dustinsfl]

Steady states for a system of nondimensionalized DEs

$$
\begin{array}{lcl}
\frac{du_1}{d\tau} & = & u_1(1 - u_1 - a_{12}u_2)\\
\frac{du_2}{d\tau} & = & \rho u_2(a - a_{21}u_1)
\end{array}
$$

So $(0,0)$ and $(1,0)$. Are there any more? If so, how did you find them?

 

[Mathwebster]

What if

$1−u_1 −a_{12} u_2 = 0$ and $a−a_{21} u_1 = 0$?

 

[Dustinsfl]

What if

$1−u_1 −a_{12} u_2 = 0$ and $a−a_{21} u_1 = 0$?

I figured it out awhile ago but thanks.