In [this MSE question/thread](https://math.stackexchange.com/questions/931782/a-conjecture-on-products-composition-of-pell-forms), I have been discussing the equation
$$
(x^2+ay^2)(u^2+bv^2) = p^2+cq^2, \tag{$\star$}
$$
where $x,a,y,u,b,v,p,c,q$ are integers. I posed a conjecture which turned out to be false. Now, as an alternative, I'm looking for a "complete integer solution" (a.k.a. integer parameterization); MO seemed like a better place to look for one, since the issue of the composition of binary quadratic forms is notoriously challenging.


Is a complete solution or parameterization already known?


If not, what's the right way to go about deriving one? I can imagine trying to resolve $x^2+ay^2=mn^2$, where [hopefully] $m$ is squarefree; doing the same for $u^2+bv^2=jk^2$ would then leave $p^2+cq^2=jm(kn)^2 = j'm'(\delta kn)^2$ where $\delta=\gcd(j,m)$, which could be resolved the same way. Is that likely the best approach?


The main application is the solution of higher-degree equations which have been reduced to the form ($\star$).



Does the Diophantine equation $(x^2+ay^2)(u^2+bv^2) = p^2+cq^2$ admit a complete solution?
As posted in [my comment above,](https://mathoverflow.net/questions/181242/does-the-diophantine-equation-x2ay2u2bv2-p2cq2-admit-a-complete#comment454532_181242) the case $a=b=c=1$ is relatively trivial to solve, using existing (nearly "classical") solutions to the 2.2.4 Diophantine sums-of-squares equation $$X\_1^2 + X\_2^2 = Y\_1^2+Y\_2^2+Y\_3^2+Y\_4^2.$$


Since nobody else was able or willing to develop a complete solution for the more general equation in the title, I am working on it myself. The complete solution will be complicated — as fully expected/anticipated — but is within reach. I will post it back here when I've fully verified my proof/algorithm.


