Parameterization of implicit equation



Hi all,

Suppose, I have an implicit equation of Cayley Cubic as below:

x^2 + y^2 - x^2*z + y^2*z + z^2 - 1 = 0

How could I rewrite it in parametric equation of x, y and z?
I have googled around about parameterization and found that there is
no definitive way to do that, rather it is a trial and error process.

Could we use Maxima to help doing parameterization?

Thank you
~ bowo ~