Finding the minimum of $x^2+y^2$ for $(x^2y-xy^2)(x^3-y^3)=x^3+y^3$ and $xy>0$. If $x,y \in \mathbb {R}$, find the minimum of  $x^2+y^2$ when $(x^2y-xy^2)(x^3-y^3)=x^3+y^3$ and $xy>0$. 
This problem was inspired by a problem which asked if $x,y \in \mathbb {R}$ and $xy \neq 0$,  find the minimum of $x^2+y^2$ when $xy(x^2-y^2)=x^2+y^2$.
By setting $x=a\sin\theta$, $y=a\cos\theta$, the equation can be simplified to $a^2=\frac{1}{\sin\theta\cos\theta(\sin\theta^2-\cos\theta^2)}$
However, notice that $\sin\theta\cos\theta=\frac{\sin2\theta}{2}$, $\sin\theta^2-\cos\theta^2=-\cos2\theta$. 
This implies that $a^2=\frac{2}{-\sin2\theta\cos2\theta}$, thus that $a^2=\frac{4}{-\sin4\theta}\ge 4$. 
Thus the minimum of $x^2+y^2$ is $4$, with the equality holding when $x=\sqrt{2-\sqrt{2}}$, $y=\sqrt{2+\sqrt{2}}$.
However, since there are no formulas I know of where $\sin^3 x+\cos^3 x$, I did not know how to find the minimum of $x^2+y^2$ when $(x^2y-xy^2)(x^3-y^3)=x^3+y^3$.
Graphing it seems to imply that such a minimum exists, but I am not aware of how to find it( and when such a minimum exists). 
Any help would be appreciated. 

Using Lagrange multiplier got a link condition: 
$$ \dfrac{x}{y} = \dfrac{x^4 y -2 x y^4 + y^5 - 3 x^2}{-2 x^4 y +5 x y^4 + x^5 -3 y^2} $$ 
