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 corresponding hierarchy bi-Hamiltonian PDE. simplest triple $(f,0,e)$ $\mathfrak{sl}_2$ corresponds KdV hierarchy, and $(f,0,e_\theta)$, where $f$ is sum negative root vectors $e_\theta$ highest vector a algebra, Drinfeld-Sokolov hierarchy.