On a class of reductions of Manakov-Santini hierarchy connected with the interpolating system

Using Lax-Sato formulation of Manakov-Santini hierarchy, we introduce a class of reductions, such that zero order reduction of this class corresponds to dKP hierarchy, and the first order reduction gives the hierarchy associated with the interpolating system introduced by Dunajski. We present Lax-Sato form of reduced hierarchy for the interpolating system and also for the reduction of arbitrary order. Similar to dKP hierarchy, Lax-Sato equations for $L$ (Lax fuction) due to the reduction split from Lax-Sato equations for $M$ (Orlov function), and the reduced hierarchy for arbitrary order of reduction is defined by Lax-Sato equations for $L$ only. Characterization of the class of reductions in terms of the dressing data is given. We also consider a waterbag reduction of the interpolating system hierarchy, which defines (1+1)-dimensional systems of hydrodynamic type.