(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
In light of the comments it's worth noting the Manin-Peyre conjecture which discusses rational points on Fano varieties. I don't see how anyone can study rational points on a manifold with no explicity formula. At least, observe that: $$ X^3 + Y^3 + Z^3 = 3 \, T^3$$ does not satisfy weak approximation yet it's a cubic surface in $\mathbb{P}^3$ and therefore Fano.
The full definition of Fano is really fancy involving the anti-canonical divisor and ample line bundles, which I doubt is how Fano or del Pezzo discussed them.
And here is more references:
- Hauteurs et Measures de Tamagawa
- An overview of Manin's conjecture for del Pezzo surfaces (many explicit examples)
