Toda frames, harmonic maps and extended Dynkin diagrams

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.