2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/148311A new approach to integrability of affine Toda field theories and closely related to them KdV hierarchies is proposed. The flows of a hierarchy are explicitly identified with infinitesimal action of the principal abelian subalgebra of the nilpotent part of the corresponding affine algebra on a homogeneous space. This is an extended version of the paper "Generalized KdV flows and nilpotent subgroups of affine Kac-Moody groups"; it has been accepted for publication in Inventiones Mathematicae.30 pages, AMSLatex; extended version accepted for publication in Invent. MathHigh Energy Physics - Theorytext