If I have a multivariate polynomial described by function $f(x,y,z)$ where $f(x,y,z)=a(x)+b(y)+c(z)$, and where $a$, $b$, and $c$ are each univariate polynomials of degree $2$, and $f(x,y,z)=0$ describes a 3-dimensional surface. Is there a mechanism for finding functions $x(u,v)$, $y(u,v)$, and $z(u,v)$ as rational functions of $u$ and $v$ such that $f(x(u,v),y(u,v),z(u,v))=0$?
What I have tried so far to solve this is explained in the paper by Chandrajit L. Bajaj on Rational Parametrizations of Nonsingular Real Cubic Surfaces. My surface is quadratic, but some other papers I have read on the subject which cite this one as a reference have indicated that the method would supposedly work for quadratics as well. However, every pair of skew lines that I pick from one particular surface I tried this with results in a denominator of $0$ in the rational parameterization formula as computed by its method, and is thus unsuitable. I would, ideally, prefer a more general system that worked for singular surfaces anyways.