2. Find all integer solutions $(x, y)$ of the following equation:
$$
7 x^{2}-40 x y+7 y^{2}=(|x-y|+2)^{3} \text {. }
$$
2. $(2,-2)$ and $(-2,2)$.

Without loss of generality, we can assume $x \geqslant y$.
At this point, the equation can be transformed into
$$
13(x+y)^{2}+(x-y-4)^{2}(2(x-y)+1)=0 \text {. }
$$

Since $2(x-y)+1>0$, we know
$$
\begin{array}{l}
x+y=0, x-y-4=0 \\
\Rightarrow x=2, y=-2 .
\end{array}
$$

Therefore, $(x, y)=(2,-2),(-2,2)$.