1. Find all integer pairs $(x, y)$ that satisfy the equation $x^{2}-2 x y+126 y^{2}=2009$. (Supplied by Zhang Pengcheng)
1. Let the integer pair $(x, y)$ satisfy the equation
$$
x^{2}-2 x y+126 y^{2}-2009=0 \text {, }
$$

considering it as a quadratic equation in $x$, its discriminant
$$
\Delta=500\left(4^{2}-y^{2}\right)+36
$$

should be a perfect square.
If $y^{2}>4^{2}$, then $\Delta<0$.
If $y^{2}<4^{2}$, then $y^{2}$ can take $0,1^{2}, 2^{2}, 3^{2}$.
The corresponding $\Delta$ values are $8036,7536,6036$, 3536, none of which are perfect squares.

Therefore, when $y^{2}=4^{2}$, $\Delta=500\left(4^{2}-y^{2}\right)+36$ $=6^{2}$ is a perfect square.
If $y=4, y=-4$, the equation (1) respectively becomes
$$
x^{2}-8 x+7=0, x^{2}+8 x+7=0 \text {. }
$$

Solving these, we get $x=1$ or $7, x=-1$ or -7.
In summary, all integer pairs satisfying the original equation are
$$
(x, y)=(1,4),(7,4),(-1,-4),(-7,-4) .
$$