  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. We give two
constructions of the integrals of these systems - an explicit construction as
limits of classical Calogero-Moser Hamiltonians of elliptic Dunkl operators as
the dynamical parameter goes to 0 (implementing an idea of
arXiv:hep-th/9403178), and a geometric construction as global sections of
sheaves of elliptic Cherednik algebras for the critical value of the twisting
parameter. We also prove algebraic integrability of these systems for values of
parameters satisfying certain integrality conditions.
