Find all solutions of this diophantine equation of the second degree in three variables

Question

Consider the Diophantine equation $Q(x,y,z)=1$, where $Q(x,y,z)$ is the quadratic form $x^2+y^2-z^2$. Let $S \subseteq {\mathbb Z}^3$ denote the set of all solutions. It is rather easy to find several parametric solutions, but it seems harder to find a complete enumeration of all the solutions. Specifically, say that a subset $T$ of ${\mathbb Z}^3$ is a polynomial image when there is a $r>0$ and three polynomial maps $P,Q,R : {\mathbb Z}^r \to {\mathbb Z}^3$ such that $T$ is the image of $(P,Q,R)$. My question is : Can $S$ be written as a finite union of polynomial images?

The older stackexchange question Integral solutions of $x^2+y^2+1=z^2$ is about the same equation, but it does not answer the specific question above.

Answer 1

Let $\Gamma$ the set of orthogonal unimodular matrices, that is $$ \Gamma=M(n,\mathbb Z) \cap O(n). $$ This group acts by left multiplication on the set $S$ of solution of your equation. Moreover, the set of $\Gamma$-orbits in $S$ is finite! That is, there is a finite set $F$ in $S$ such that: for any $X\in S$ there are $g\in\Gamma$ and $Y\in F$ such that $$ X=gY. $$ I think this allows you to write $S$ as a finite union of polynomial images. Am I right?