Remarks on intersection numbers and integrable hierarchies. II. Tau-structure

For systems of evolutionary partial differential equations (PDEs), the tau-structure is an important notion that originated from the deep relation between integrable systems and quantum field theories. We show that, under a certain non-degeneracy condition, existence of a tau-structure implies integrability. As an example, we apply this principle to provide a new proof of the integrability of the Drinfeld–Sokolov (DS) hierarchy associated with an arbitrary Kac–Moody algebra and a choice of a vertex of its Dynkin diagram.