Find parametrization of a manifold Find parametrization of $x^2y^2+y^2z^2+z^2x^2=xyz$, $x,y,z>0$.
I've checked it's manifold but I can't find parametrization(s).
 A: So, $$\frac {xy}z+\frac{yz}x+\frac{zx}y=1$$
As $x,y,z>0$ 
we can write   $\frac {xy}z+\frac {yz}x=\cos^2\theta$ and $\frac{zx}y=\sin^2\theta$
So,  $\frac {xy}{z\cos^2\theta}+\frac {yz}{x\cos^2\theta}=1$ 
So, we can write  $\frac {xy}{z\cos^2\theta}=\cos^2\phi$ and $\frac {yz}{x\cos^2\theta}=\sin^2\phi$ as $\frac {xy}{z\cos^2\theta},\frac {yz}{x\cos^2\theta}>0$ 
So,  $\frac {xy}z=\cos^2\theta\cos^2\phi--->(1)$  
$\frac {yz}x=\cos^2\theta\sin^2\phi--->(2)$  
$\frac{zx}y=\sin^2\theta--->(3)$
On multiplication, $xyz=\cos^4\theta\cos^2\phi \sin^2\phi \sin^2\theta-->(4) $ 
$(4)/(1),z^2=\cos^2\theta\sin^2\phi \sin^2\theta\implies z=\cos\theta\sin\phi \sin\theta=\frac{\sin\phi \sin2\theta}2$
Similarly, $(4)/(2)$ and $(4)/(3)$ will supply 
$x=\frac{\cos\phi \sin2\theta}2$ and $y=\cos\phi\sin\phi \cos^2\theta$
As $x,y,z>0$
either $\sin2\theta, \sin\phi,\cos\phi>0\implies 0<\phi<\frac{\pi}2$ and $0<2\theta<\pi$ with $\theta\ne \frac{\pi}2$
or, $\sin2\theta, \sin\phi,\cos\phi<0\implies \pi<\phi<\frac{3\pi}2$ and $\pi<2\theta<2\pi$ with $\theta\ne \frac{3\pi}2$
