Do there exist integers $x$ and $y$ such that $\frac{x^3-x^2-2 x+1}{y^2-4}$ is an integer?

In other words, can any integer representable as $x^3-x^2-2 x+1$ have any divisor representable as $y^2-4$?

Do there exist integers $x$ and $y$ such that $\frac{x^3-x^2-2 x+1}{y^2-4}$ is an integer?

In other words, can any integer representable as $x^3-x^2-2 x+1$ have any divisor representable as $y^2-4$?

This is the simplest non-trivial example of my earlier question Integer points of rational function in 2 variables .