Checking solvability of BDQE

Question

Consider a BDQE:

$$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 $$

where $ A,B,C,D,E,F \in \mathbb Z$

Is there a method to determine (prove/disprove) if integer solution(s) to this equation exist(s) without actually calculating it/them?

Answer 1

Ok, the general problem of solvability of Diophantine equations (not only BDQE, all of them) is the content of 10th Hilbert's Problem which was solved with a negative answer (no general algorithm exists). Still, I thought there could be an exception for specific subclasses of DE like BDQE, but again this is not true.