Find all integers $(x, y)$ such that $2 x^{3}+x y-7=0$.
We write $x\left(2 x^{2}+y\right)=7$. It follows that $\left(x, 2 x^{2}+y\right)= \pm(1,7)$ or $\pm(7,1)$. We test the different possible values of $x$ and obtain the solutions: $(-7,-99),(-1,-9),(1,5),(7,-97)$.