Graded Lie algebras, representation theory, integrable mappings and systems. Non-abelian case Original Research Article

The exactly integrable systems connected with a semisimple series A for arbitrary grading are presented in an explicit form. Their general solutions are expressed in terms of the matrix elements of various fundamental representations of the An groups. The simplest example of such systems is the generalized Toda chain with matrices of arbitrary dimensions in each point of the lattice.