  In this paper we introduce the notion of coalgebra symmetry for discrete
systems. With this concept we prove that all discrete radially symmetric
systems in standard form are quasi-integrable and that all variational discrete
quasi-radially symmetric systems in standard form are
Poincar\'e--Lyapunov--Nekhoroshev maps of order $N-2$, where $N$ are the
degrees of freedom of the system. We also discuss the integrability properties
of several vector systems which are generalisations of well-known one degree of
freedom discrete integrable systems, including two $N$ degrees of freedom
autonomous discrete Painlev\'e I equations and an $N$ degrees of freedom
McMillan map.
