Given a vector $v\in\mathbb Z^n$ and pairwise commutative matrices $M_1,\dotsc,M_k\in \operatorname{SL}(n,\mathbb Z)$, how to compute all 01-vectors in the orbit of $v$ with respect to multiplication by elements of $\langle M_1,\dotsc,M_k\rangle$?
This question is inspired by a particular approach to the previous one: Recover cyclotomic integer with bounded coefficients from its known associate.