2. In the set of integers, solve the equation

$$
x^{2}+x y+y^{2}=x^{2} y^{2}
$$
Solution. The given equation is equivalent to the equation

$$
(x+y)^{2}=x y(x y+1)
$$

The product of two consecutive integers is a square of an integer if and only if those integers are $-1,0$ or 0,1. Therefore, it must be that $x y=-1$ or $x y=0$ and in both cases $x+y=0$. Solving the corresponding systems easily yields that the only possible solutions to the given equation are: $(0,0),(1,-1),(-1,1)$. That these are indeed solutions to the equation can be verified by direct substitution.