Vector fields and invariants of the full symmetric Toda system

The geometric properties of the full symmetric Toda systems are studied. A simple geometric construction is described that allows constructing a commutative family of vector fields on a compact group including the Toda vector field, i.e., the field that generates the full symmetric Toda system associated with the Cartan decomposition of a semisimple Lie algebra. Our construction involves representations of a semisimple algebra and is independent of whether the Cartan pair is split. The result is closely related to the family of invariants and semiinvariants for the Toda system on $SL_n$.