= Solution
The <fixed point> equations give $x=\mu y$ and
$$
0=-\mu x+\mu y+y^3=y\bigl(y^2+\mu-\mu^2\bigr).
$$
Thus the origin exists for every real $\mu$. The nonzero <fixed points> are
$$
\boxed{(x,y)=\left(\pm\mu\sqrt{\mu(\mu-1)},\ \pm\sqrt{\mu(\mu-1)}\right),\qquad \mu<0\ \text{or}\ \mu>1,}
$$
with the same sign in both positions. At $\mu=0$ or $\mu=1$ these expressions coalesce with the origin and do not give additional <fixed points>. There are no nonzero real <fixed points> for $0<\mu<1$.
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