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Compute 01-vectors in the orbit of a given vector wrt a finitely-generated abelian subgroup of SL(n,ℤ)

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 ...