# Minimal bivariate diophantine equation solution space

I am facing the following type of diophantine equations:

$$axy + bx + cy + d = 0$$

Where $a$, $b$, $c$, $d$ are integers and solutions for $x$, $y$ in the integers are seeked. If $a=0$ one can apply the extended euclidean algorithm. $x$ and $y$ can then be viewed as generated by a new parameter $t$ and linear forms:

$$x = et + f \\ y = gt + h$$

Can something similar be said when $a\ne0$?

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$axy+bx+cy+d=0$; $a^2xy+abx+acy+ad=0$; $(ax+c)(ay+b)+ad-bc=0$; $(ax+c)(ay+b)=bc-ad$. So what you have to do is factor $bc-ad$; then for every factorization $bc-ad=rs$, you have to see whether the simultaneous equations $ax+c=r$, $ay+b=s$ have a solution.