Coalgebra symmetry for discrete systems

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