To every irreducible finite crystallographic reflection group (i.e., an
irreducible finite reflection group G acting faithfully on an abelian variety
X), we attach a family of classical and quantum integrable systems on X (with
meromorphic coefficients). These families are parametrized by G-invariant
functions of pairs (T,s), where T is a hypertorus in X (of codimension 1), and
s in G is a reflection acting trivially on T. If G is a real reflection group,
these families reduce to the known generalizations of elliptic Calogero-Moser
systems, but in the non-real case they appear to be new.