The following question arises, for me, from mathematical music theory:

Write $({\Bbb Z}^n,E_n)$ for the Cayley graph of ${\Bbb Z}^n$ relative to standard free generators.

Given a subgroup $L$ of ${\Bbb Z}^n$ of finite index, how ought one efficiently count and/or enumerate, up to translation, the connected (as vertex-induced subgraphs) fundamental domains for the action of $L$ on $({\Bbb Z}^n,E_n)$.