Part of this is finding the primitive integer null vectors of an indefinite ternary quadratic form $f(x,y,z).$ Mordell points out that these occur in a finite number of parametrizations. The process you mention, stereographic projection around a rational point, does not do a good job of finding primitive integral solutions. Instead, it immediately finds all rational solutions, with no bound on denominators.

The Hessian matrix $H$ of an isotropic ternary form has this feature: there is an integer matrix $P$ and an integer $n$ such that $$ P^T HP = nG \; , $$ where $G$ is the Hessian matrix of $g(x,y,z) = y^2 - zx \; . \;$ Indeed, there are infinitely of these. For a fixed $n,$ there are typically several such $P$ if any.

Let's see, the primitive null vectors of $y^2 - zx$ are precisely $(p^2,pq,q^2).$ Applying $P$ to this (as a column vector) gives a null vector for $H,$ and we get some ability to say when this will be primitive.

I worked this out for isotropic forms of the sort $A(x^2 + y^2 + z^2) - B(yz+zx+xy).$ The number of inequivalent $P$ matrices needed to produce all primitive null vectors can be arbitrarily large. I kept a list somewhere...

The original proof is in Fricke and Klein (1897), where it is mentioned in passing. Different versions have been published over the years. I eventually wrote down a proof using just matrices, gcd and the like.