When does $A(x^2+y^2+z^2)=B(xy + yz + xz)$ have nontrivial integer solutions? Someone on MathematicaSE asked for which coprime integer pairs $(A,B)$ satisfying $A<B$ the equation $A(x^2+y^2+z^2)=B(xy + yz + xz)$ admits nontrivial integer solutions. The question was closed there, but I figured it would be a nice courtesy to re-ask it here instead. 
There are really three questions:


*

*Are all such $(A,B)$ pairs known?

*If not, is there any reference containing an exhaustive list of all pairs satisfying $0<A<B<N$ for some moderately large integer $N$ (say $N=1000$)?

*If not, does there exist a software library or algorithms which provide an efficient search method that avoids brute-force enumeration?

 A: If you know that the issue was previously discussed. It is not clear what meaning to ask the same question?
Find a solution: $3(x^2+y^2+z^2)=10(xy+yz+zx)$
Solutions to $ax^2 + by^2 = cz^2$
http://www.artofproblemsolving.com/blog/98932
http://www.artofproblemsolving.com/blog/98931
http://www.artofproblemsolving.com/blog/98930
http://www.artofproblemsolving.com/blog/98929
If these formulas are not satisfied, then we must ask the question more specifically.  What kind of decisions need to look for?
I just can not understand.  A number of different formulas can be written for this equation. Depending on how the coefficients should look, but still stubbornly use the computer to search solutions.
Can this have any meaning? I don't understand! I can explain what is the meaning?
A: One of the solutions to the equation. To get infinite amount of different.
$$A(x^2+y^2+z^2)=B(xy+xz+yz)$$
If you can imagine.    $$A=ps$$  $$B=p^2+s^2$$
Then decisions can be recorded.
$$x=pt^2-(p^2+s^2)tk+s(p^2-ps+s^2)k^2$$
$$y=st^2-(p^2+s^2)tk+p(p^2-ps+s^2)k^2$$
$$z=tk(p-s)^2$$
If you can imagine.  $$A=s(s-2p)$$  $$B=p^2+2s^2$$
Then decisions can be recorded.
$$x=(s-2p)t^2+2(p+s)tk+sk^2$$
$$y=(s-2p)t^2-2(p+s)tk+sk^2$$
$$z=(p+4s)t^2-pk^2$$
$t,k$ - integers which we ask.
