I consider the surface of degree $5$ of equation: $$ 4 x y - 4 x^3 y - 4 x y^3 + z + 2 x^2 z + x^4 z + 2 y^2 z + 2 x^2 y^2 z + y^4 z - 4 x y z^2 - 6 z^3 + 2 x^2 z^3 + 2 y^2 z^3 + z^5=0 $$
Question: is this surface rational -- is there a parametrization for this surface using rational functions of two variables?
A more general question would be whether there is a parametrization using some special functions.
Thanks, any help would be appreciated!