Školjka

Let $2\mathbb{Z} + 1$ denote the set of odd integers. Find all functions $f:\mathbb{Z} \mapsto 2\mathbb{Z} + 1$ satisfying \[ f(x + f(x) + y) + f(x - f(x) - y) = f(x+y) + f(x-y) \]for every $x, y \in \mathbb{Z}$. \begin{flushright}\emph{(U.S.A.)}\end{flushright}