I'm intrested in studying integer points on quadratic equations:
$$Q(x,y)=ax^2+bxy+cy^2+dx+ey+f=0, \quad (a,b,c,d,e,f\in\mathbb{Z}).$$
Using Lagrange method it is possible to reduce it in a Pell equation
$$X^2-DY^2=N,$$
with
$$ X=(b^2-4ac)y+E, \quad Y=2ax+by+d,\quad N=E^2-F(b^2-4ac), $$
easy to solve algorithmically if $Q(x,y)$ is not a parabola $(\Delta=b^2-4ac\neq 0)$.
How can I determine integer points on a quadratic equation with $\Delta =0$?
If $b^2-4ac=0$ and $a,b,c$ are not all zero, then you first transform the equation into $$ A(\alpha x+\beta y)^2+B(\alpha x+\beta y)+C(\gamma x+\delta y)+D=0, \begin{pmatrix}\alpha&\beta\\\gamma&\delta\end{pmatrix}\in SL(2,\mathbb{Z}) $$ and hence $AX^2+BX+CY+D=0$. Completing the square, you can further reduce it to the form $A^2X^2+CY+D=0$. Then it is basically constraining $X$ by $(AX)^2\equiv -D\pmod{C}$ (which may or may not have any solutions $X_0$) and thus for each $X=Cm+X_0$ ($m\in\mathbb{Z}$, $(AX_0)^2\equiv-D\pmod{C}$), obtain the resulting $Y$.