  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.
