Dispersionful analogues of Benney's equations and N-wave systems

We recall Krichever's construction of additional flows to Benney's hierarchy, attached to poles at finite distance of the Lax operator. Then we construct a 'dispersionful' analogue of this hierarchy, in which the role of poles at finite distance is played by Miura fields. We connect this hierarchy with N-wave systems, and prove several facts about the latter (Lax representation, Chern-Simons-type Lagrangian, connection with Liouville equation, tau-functions).