If $X$ is the set of all points $(x,y,z)$ such that $(x^{2}+y^{2}+z^{2})^{2} = 4(x^{2}+y^{2})$, at what points will $X$ be a smooth surface? From what I understand, $X$ is a hollow ellipsoid (with $x$-$y$ radius $= 2$), and the top and bottom of this ellipsoid are caving toward the point at the origin. This would make me say that $X$ is smooth for all $(x,y,z)$ in $X$ such that $(x,y,z)$ is not $(0,0,0)$.
See both http://www.wolframalpha.com/input/?i=%7B-sqrt(-x%5E2+-+y%5E2+%2B+2+sqrt(x%5E2+%2B+y%5E2)),+y%3D+-2+to+2%7D and http://www.wolframalpha.com/input/?i=%7Bsqrt(-x%5E2+-+y%5E2+%2B+2+sqrt(x%5E2+%2B+y%5E2)),+y%3D+-2+to+2%7D for illustrations of $X$.