Let given $p,q \in \mathbb{Z}^+,r \in \mathbb{Z}$ and equation $p x^2 + q y^2 = z^3 + r$.
If exist solutions of Pell equation $qb^2-3pa^2=r\pm1$, then
$x = a(pa^2 - 3qb^2 + 3r)\\ y = b\\z = pa^2 + qb^2 - r$
Exist other Pell-kind parametrization of source equation with more symmetrical formulas for $x,y$?
The context of the question is related to the sum of cubes problem.
$u^3+v^3-z^3=r \implies u^3+v^3=z^3+r=px^2+qy^2$
For known $r$ and "big" $z$ and unknown $x,y$ equation $z^3+r=px^2+qy^2$ is always solvable for some "small" $p,q$.
For known $p,q,r$ and unknown $x,y,z$ equation $z^3+r=px^2+qy^2$ is solvable by Pell-like parametrization.
For known $z$ validity check $u,v$ is fast.
