The exactly integrable systems connected with semisimple series $A$ for arbitrary grading are presented in explicit form. Their general solutions are expressed in terms of the matrix elements of various fundamental representations of $A_n$ groups. The simplest example of such systems is the generalized Toda chain with the matrices of arbitrary dimensions in each point of the lattice.
Graded Lie algebras, representation theory, integrable mappings and systems
Integrable mappings and the notion of anticonfinement
LieART -- A Mathematica Application for Lie Algebras and Representation Theory
