Differential geometry of Toda systems (Q1904287)

The grading condition met in the group-algebraic approach of integrable systems presented by \textit{A. N. Leznov} and the second author [Group-theoretical methods for integration of nonlinear dynamical systems, Birkhäuser-Verlag, Basel (1992)] receives now a differential geometry formulation in terms of some holomorphic distributions on a certain flag manifold. The derivation of the related nonlinear systems is considered. For the simple Lie groups one comes to the Toda systems. The reality condition for their solutions is introduced. Two generalized Plücker representations are found. These lead to some formulas involving Toda fields viewed as Kähler potentials. Two appendices on geometry of complex Lie groups and algebras make more accessible this very interesting paper.