  The purpose of this paper is to connect two subjects: the theory of quantum
integrable systems (complete commutative rings of differential operators), and
differential Galois theory. We define quantum completely integrable systems
(QCIS), algebraically integrable QCIS, the differential Galois group of a QCIS.
We show that the differential Galois group is always reductive and that a QCIS
is algebraically integrable if and only if its differential Galois group is
commutative. In particular, we show that a differential operator L in one
variable is algebraic in the sense of Krichever (i.e. finite-zone) if and only
if the differential Galois group of the differential equation Lf=af is
commutative for a generic number a. As a by-product, we obtain a proof of the
Veselov-Chalyh conjecture on the algebraic integrability of the elliptic
Calogero-Moser system.
