<s>[INST] Solve $x^2y^2 - 4x^2y + y^3 + 4x^2 - 3y^2 + 1 = 0$ over the integers. 
Solve $$x^2y^2 - 4x^2y + y^3 + 4x^2 - 3y^2 + 1 = 0$$ over the integers.

You can probably guess by now... This problem is adapted from a recent competition.
If there are any other solutions, please post them below. I have provided one if you want to check out.
 [/INST] Write $$x^2(y-2)^2 = -y^3+3y^2-1$$
Since $y-2\mid -y^3+3y^2-1$ and $y\equiv 2\pmod{y-2}$ we have
$$0\equiv -y^3+3y^2-1 \equiv -8+12-1 \equiv 3 \pmod{y-2}$$   So $$y-2\mid 3\implies y-2\in\{1,-1,3,-3\}$$
so $$y\in\{3,1,5,-1\}$$
Checking each of them we are done. 
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