II. (50 points) Find all integer solutions to the equation $x^{2}-2009 y+2 y^{2}=0$.

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II. (50 points) Find all integer solutions to the equation $x^{2}-2009 y+2 y^{2}=0$.
II. Sol: $\because x^{2}+2 y^{2}=2009 y=7^{2} \times 41 y, \therefore x^{2}+2 y^{2} \equiv 0(\bmod 7)$. Let $x=7^{2} \times a, y=7^{2} \times b, a, b \in \mathbf{Z}$, the original equation can be transformed into: $a^{2}+2 b^{2}=41 b$. $\therefore a^{2}=(41-2 b) b, \therefore 41-2 b$ and $b$ are both perfect squares.
$\therefore$ After inspection, we can get $b=0$ or $b=16$.
$\therefore$ The solutions $(x, y)$ of the original equation are: $(0,0) 、( \pm 588,784)$.