A class of multidimensional integrable hierarchies and their reductions

We consider a class of multidimensional integrable hierarchies connected with the commutativity of general (unreduced) $(N+1)$-dimensional vector fields containing a derivative with respect to a spectral variable. These hierarchies are determined by a generating equation, equivalent to the Lax–Sato form. We present a dressing scheme based on a nonlinear vector Riemann problem for this class. As characteristic examples, we consider the hierarchies connected with the Manakov–Santini equation and the Dunajski system.