(unusual move to answer my own question)
In fact this is duplicate:
and the rational points are of course easy to find
$$ x=\frac{2u}{u^2+v^2+1};y=\frac{2v}{u^2+v^2+1};z=\frac{u^2+v^2-1}{u^2+v^2+1} $$
There is even a proof of equidistribution as a result of Duke He shows that:
$$ \frac{\sum_{h(x)\leq T} \psi(x)}{\sum_{h(x)\leq T} \;\;1\;\;} \to \int_{S^2} \psi \, d\mu $$ for any function $\psi: S^2 \to \mathbb{C}$. There is also general discussion of Bjorn Poonen in genral on rational points on varieties