They're Not All The Same Degree!

Number Theory Level 5

$\large \dfrac {x^3+y^3}{3}+xy = 2007$

Let there be $n$ unordered positive integral solutions to the above equation. Call these solutions $(x_1, y_1), (x_2, y_2) , \ldots, (x_n, y_n)$ . Find the value of

$\displaystyle \sum_{i=1}^n x_i y_i.$