Find stationary points of the function $f(x,y) = (y^2-x^4)(x^2+y^2-20)$ I have problem in finding some of the stationary points of the function above. I proceeded in this way: the gradient of the function is:
$$ \nabla f = \left( xy^2-3x^5-2x^3y^2+40x^3 ; x^2y+2y^3-x^4y-20y \right) $$
So in order to find the stationary points, I must resolve the system:
$$ \begin{cases} 
xy^2-3x^5-2x^3y^2+40x^3 = 0 \\
x^2y+2y^3-x^4y-20y = 0
\end{cases} $$
So far I've found the points:
$$ (0,0) \qquad \left(\pm 2 \sqrt{10 \over 3} , 0 \right) \qquad (0, \pm \sqrt{10}) $$
But, I'm still blocked  when I've to found the points deriving by the system:
$$
\begin{cases} 
2x^6 + 3x^4 +x^2 -20 = 0 \\
y^2 = \frac{1}{2} \left( x^4 - x^2 + 20 \right) \end{cases}
$$
Which I don't know how to solve. Can someone help me ? Thanks.

WA gets
$$
\DeclareMathOperator{grad}{grad}
\grad((x^2+y^2-20) (y^2-x^4)) = (-6 x^5-4 x^3 (y^2-20)+2 x y^2, 2 y (-x^4+x^2+2 y^2-20))
$$
(link) and nine real critical points (link).

