\(x^2-2x+2y=2\left(xy+2\right)\) \(\left(x;y\in Z\right)\)
\(\Leftrightarrow x^2-2x+1+2y-2xy=5\)
\(\Leftrightarrow\left(x-1\right)^2-2y\left(x-1\right)=5\)
\(\Leftrightarrow\left(x-1\right)\left(x-2y-1\right)=5\)
\(\Leftrightarrow\left(x-1\right);\left(x-2y-1\right)\in U\left(5\right)=\left\{-1;1-5;5\right\}\)
\(\Leftrightarrow\left(x;y\right)=\left(6;2\right)\)