  Let $K$ and $L$ be origin-symmetric convex integer polytopes in
$\mathbb{R}^n$. We study a discrete analogue of the Aleksandrov projection
problem. If for every $u\in \mathbb{Z}^n$, the sets $(K\cap
\mathbb{Z}^n)|u^\perp$ and $(L\cap \mathbb{Z}^n)|u^\perp$ have the same number
of points, is then $K=L$? We give a positive answer to this problem in
$\mathbb{Z}^2$ under an additional hypothesis that $(2K\cap
\mathbb{Z}^2)|u^\perp$ and $(2L\cap \mathbb{Z}^2)|u^\perp$ have the same number
of points.
