  We prove that all immersions of a genus one surface into G/T possessing a
Toda frame can be constructed by integrating a pair of commuting vector fields
on a finite dimensional Lie algebra. Here G is any simple real Lie group (not
necessarily compact), T is a Cartan subgroup and the k-symmetric space
structure on G/T is induced from the Coxeter automorphism. We provide necessary
and sufficient conditions for the existence of a Toda frame for a harmonic map
into G/T and describe those G/T to which the theory applies in terms of
involutions of extended Dynkin diagrams.
