Is it known exactly for which integers $a,b,c$ the equation $ax^2+bxy+cy^2=0$ has a nontrivial solution? If not, what is known about this problem in general? (I am asking about the specific problem of "for which integers $a,b,c$ does $ax^2+bxy+cy^2=0$ have a nontrivial solution in integers $x$ and $y$?)
Thanks