<s>[INST] Diophantine Equation: $xy+ax+by+c=0$ How to find integer solutions $x,y$ of $xy+ax+by+c=0$ for given $a,b,c \in \mathbb{Z}$?
Is there somewhere a treatise on this kind of equations? 
 [/INST] (I'm guessing that what was intended was $xy+ax+by+c=0$.)
\begin{align}
u & = x+y \\
v & = x-y \\[8pt]
\text{so that }x & = \frac{u+v}{2} \\[8pt]
y & = \frac{u-v}{2}
\end{align}
Then $xy = \dfrac{u^2-v^2}{4}$.  So
$$
xy + ax + by + c =0
$$
becomes
$$
\frac{u^2-v^2}{4} + a\frac{u+v}{2} + b \frac{u-v}{2} + c =0,
$$
and then multiplying both sides by $4$, we get
$$
u^2 - v^2 + 2(a+b)u + 2(a-b)v + c = 0.
$$
Then one can complete the square, getting
$$
(u+a+b)^2 - (v-a+b)^2 = (a+b)^2+(a-b)^2 - c = 2a^2+2b^2-c.
$$
</s>