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 $a\ne0$?