$\mathrm{e}^{2x+y+z}=x^2+y^2-2x-4y+4$ is a rotation surface?

Question

$\mathrm{e}^{2x+y+z}=x^2+y^2-2x-4y+4$ is a rotation surface?

Why?

1. By translation, $e^{2u+v+w+4}=u^2+v^2-1$.

2. By orthogonal transform as $u=(2x+y+z)/6^{1/2}$, or $u=(2x+y)/5^{1/2}$, we do not find ...

3. Which translation and orthogonal transform should be put so that the surface becomes a rotation surface around $z$ axis?

4. Or what can be said easily why it is a rotation surface? Oh...

Answer 1

Here is a plot. It doesn't look like a surface of rotation.