One, (20 points) Find all positive integer solutions $x, y$ that satisfy the equation
$$
x^{2}+y^{2}=2(x+y)+x y
$$
$$
\begin{array}{l}
x^{2}-(y+2) x+y^{2}-2 y=0 . \\
\text { By } \Delta=(y+2)^{2}-4\left(y^{2}-2 y\right) \\
=-3 y^{2}+12 y+4 \\
=16-3(y-2)^{2} \geqslant 0 \\
\Rightarrow(y-2)^{2} \leqslant \frac{16}{3}<9 \\
\Rightarrow-3<y-2<3 \\
\Rightarrow-1<y<5 .
\end{array}
$$

Since $y$ is a positive integer, thus,
$$
y=1,2,3,4 \text {. }
$$

Therefore, the positive integer solutions of the equation are
$$
\left\{\begin{array} { l } 
{ x = 2 , } \\
{ y = 4 }
\end{array} \text { or } \left\{\begin{array} { l } 
{ x = 4 , } \\
{ y = 2 }
\end{array} \text { or } \left\{\begin{array}{l}
x=4, \\
y=4 .
\end{array}\right.\right.\right.
$$