Find the equilibria of the following system of ODEs 
Find the equilibria of the following system of ODEs
\begin{align*}
  \dot{x} = ax - y + kx(x^2 + y^2)\\
  \dot{y} = x - ay + ky(x^2 + y^2)
  \end{align*}
where $a$ and $k$ are constants, $a > 1$ and $a^2 \geq 1$. 

I want to find the equilibria of this system and say something about the behaviour in the phase-plane. Unfortunately, I can't seem to find the equilibria.
What I've tried: I subtracted the second equation from the first one to arrive at
\begin{align}
ax + ay -y -x+kx(x^2 + y^2) - ky(x^2 + y^2) = 0 \\
\Leftrightarrow a(x + y) - (x+y) + (x - y)k(x^2 + y^2) = 0\\
\Leftrightarrow a - 1 + \dfrac{x - y}{x+y}(x^2 + y^2)k = 0
\end{align}
From here I want to find an expression for $x$ or $y$ but I don't know how to do so. 
Question: How should I approach this? If I'd get a hint on how to proceed I think I can solve the problem myself.
Thanks!

Set $\dot x=\dot y=0$. Multiply the first equation with $x$, the second with $y$ and add to get
$$
0=a(x^2-y^2)+k(x^2+y^2)^2.
$$
Now do everything askew, multiply the first equation with $y$, the second with $x$ and subtract to get
$$
0=2axy-(x^2+y^2)
$$
Apart from the origin, there are also solutions for $y=q_\pm x=(a\pm\sqrt{a^2-1})x$, and from the first equation,
$$
y^2-x^2=4kax^2y^2\implies q^2-1=4kaq^2x^2,~~ x^2=\frac{q^2-1}{4kaq^2}
$$
which only gives real solutions for $q=q_+=a+\sqrt{a^2-1}>1$.

Example with $a=2$, $k=3$ using WolframAlpha
streamplot[{2x-y+3x(x^2+y^2), x-2y+3y(x^2+y^2)}, {x,-1.5,1.5}, {y,-1.5,1.5}]


showing a saddle point at the origin and two additional centers at the other two stationary points. At the boundary of the plot all solutions point outwards.
