Let $x,y$ be integers, and $f(x,y)$, $g(x,y)$, and $h(x,y)$ be polynomials in $x$ and $y$ with integer coefficients such that $$ f(x,y)^2 - g(x,y)h(x,y)^2 = 1. \qquad(\star) $$ Furthermore, assume it can be proven that $g(x,y)$ is a (possibly unknown) non-square integer $d$, so that we can write $(u,v)=\left(f(x,y),h(x,y)\right)$ and $(\star)$ becomes the Pell equation $u^2-dv^2=1$.
What can now be said about the components of $(\star)$? Evidently, we have the easy conclusions that $f(x,y)$, $g(x,y)$, and $h(x,y)$ are pairwise relatively prime, and that $f(x,y)/h(x,y) \approx \sqrt{g(x,y)}$.
Is there anything else substantive we can deduce in general? Does the additional restriction $\gcd(x,y)=1$ allow anything further to be deduced?
EDIT: Here are a couple of interesting resources/papers dealing with special cases of $(\star)$.
Nathanson http://www.jstor.org/discover/10.2307/2041581?uid=3739448&uid=2&uid=3737720&uid=4&sid=21104074640901
Yakota http://www.sciencedirect.com/science/article/pii/S0022314X10000363 and http://www.sciencedirect.com/science/article/pii/S0022314X04000228