All pairs (x,y) that satisfy the equation $xy+(x^3+y^3)/3=2007$ How we can find the all pairs $(x,y)$  from the integers numbers ,that satisfy the equation :
$$xy+\frac{x^3+y^3}{3} =2007$$

Note that the given equation is $$x^3+y^3+3xy=6021$$ or $$x^3+y^3-1+3xy=6020$$
Factoring it we get , $$(x+y-1)(x^2+y^2+1+x+y-xy)=2^2.5.7.43$$
Obviously check $\equiv 3$ and see $$x+y-1\equiv 2 \mod 3$$ 
Also, $$x+y-1 < x^2+y^2+1+x+y-xy$$ 
This means, $$x+y-1 \rightarrow 20,5,2,35$$ 
so now it's easy to see that $$x+y-1=20$$ and so $(x, y)=(18,3)$ or $(3,18)$
