  A 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.
