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I need an implicit function that plots the surface that I am showing you in the picture. Everything you need is shown there. The surface is a tube in the shape of a parabola. The radius of its cross-sections is $3$.

parabola

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    A parabola shaped toroidal tube. If not ok, I can put back old title before edit.2015-08-18

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Rotating can be done by a "simple" substitution of variables.

This was just wrong! Rotating 45 degrees around the $z$-axis is done by replacing: $x$ with $\sqrt{(x^2+y^2)}$ and $y$ with $\sqrt{(x^2-y^2)}$ (that might be 45 degrees in the wrong direction for your case, but in that case swap $+$ and $-$).

The substitution should describe the inverse transformation of what you want (you can do much more than rotations with variable substitutions), so for a rotation of 45 degrees around the $z$-axis is (I hope I get it right this time):
$x=\frac{1}{\sqrt{2}}x-\frac{1}{\sqrt{2}}y$ and
$y=\frac{1}{\sqrt{2}}x+\frac{1}{\sqrt{2}}y$.
$\left(\frac{1}{\sqrt{2}}=\cos(\frac{\pi}{2})=\sin(\frac{\pi}{2})\right)$

It won't make your equation any prettier though...