  In this paper we study the relationship between the integrability of rational
symplectic maps and difference Galois theory. We present a Galoisian condition,
of Morales-Ramis type, ensuring the non-integrability of a rational symplectic
map in the non-commutative sense (Mishchenko-Fomenko). As a particular case, we
obtain a com- plete discrete analogue of Morales-Ramis Theorems for
non-integrabi- lity in the sense of Liouville.
