Answer: $x=1, y=2$.

Solution. This equation can be transformed into

$$
\begin{gathered}
x^{2}-2 x+5=4 y-y^{2}-|y-2 x|, \quad x y \neq 0 \\
(x-1)^{2}+(y-2)^{2}+|y-2 x|=0
\end{gathered}
$$

Each term on the left is non-negative, so the sum will be zero only if each term is zero: $x-1=0, y-2=0, y-2 x=0, x y \neq 0$. From this, we obtain the answer.
1. Find all solutions of the equation

$$
\frac{x-2}{y}+\frac{5}{x y}=\frac{4-y}{x}-\frac{|y-2 x|}{x y}
$$