Integrable triples in semisimple Lie algebras

We classify all integrable triples in simple Lie algebras, up to equivalence. The importance of this problem stems from the fact that for each such equivalence class one can construct the corresponding integrable hierarchy of bi-Hamiltonian PDE. The simplest integrable triple (f,0,e) in sl2 corresponds to the KdV hierarchy, and the triple (f,0,eθ), where f is the sum of negative simple root vectors and eθ is the highest root vector of a simple Lie algebra, corresponds to the Drinfeld-Sokolov hierarchy