Аннотация:In this paper we continue our study of the geometric properties of full symmetric
Toda systems from [CSS14,CSS17,CSS19]. Namely we describe here a simple
geometric construction of a commutative family of vector fields on compact groups,
that include the Toda vector field, i.e. the field, which generates the full symmetric
Toda system associated with the Cartan decomposition of a semisimple Lie algebra.
Our construction makes use of the representations of the semisimple algebra and
does not depend on the splitness of the Cartan pair. It is very close to the family of
invariants and semiinvariants of the Toda system associted with SLn, introduced
in [CS].