Solving Quadratic Diophantine Equations with A=0, C=0

[sparsh12]

A quadratic diophantine equation is of form:

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

Now, for A=0 and C=0,

Bxy + Dx + Ey + F=0 ...(1)

moreover there is one more condition, gcd(B,D,E)=1

So how do I find if some integral solution of (1) exists or not?
I am not interested in the solution itself, but rather just it's existence.

And the method must not depend on searching, as in the image here:
http://s9.postimage.org/dv30vaixb/diop.png

Original website was:
http://www.alpertron.com.ar/METHODS.HTM#SHyperb

Thanks in advance for advice and ideas.

 

[sparsh12]

Why no reply yet?

 

[DonAntonio]

Well, it's very hard to know what EXACTLY you want, as you say that you want to know about the existence of an integral solution "without searching" (??), but in many instance one HAS to divide the problem in cases and check each, something you apparently don't want to do...

 

[coolul007]

The equation is the standard equation for all conic sections in 2 dimensions. It seems very plausible that there are circles/parabolas/hyperbolas that have integer roots.