= Solution
A <stationary point> must satisfy both factored equations from part (a). If either coordinate is zero, the other equation forces the other coordinate to vanish. This gives $(0,0)$.
Otherwise $x^4=8y^2$ and $y^4=8x^2$. With $u=x^2>0$ and $v=y^2>0$, these become $u^2=8v$ and $v^2=8u$. Dividing gives $(u/v)^3=1$, so $u=v$. Then $u^2=8u$ forces $u=v=8$. Hence all five <stationary points> are
$$
\boxed{(0,0),\qquad(2\sqrt2\,\varepsilon,\ 2\sqrt2\,\delta),\quad \varepsilon,\delta\in\{-1,1\}}.
$$
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