MathNet 1997 — Part 01 Question 43

Let $\mathbb{Z}$ denote the set of integers. Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that $f(-1) = f(1)$ and $f(x) + f(y) = f(x + 2xy) + f(y - 2xy)$ for all integers $x, y$.