Interpolating differential reductions of multidimensional integrable hierarchies

We apply the scheme for constructing differential reductions recently developed for the Manakov–Santini hierarchy to the general multidimensional case. We consider the four-dimensional case connected with the second heavenly equation and its generalization proposed by Dunajski in more detail. We characterize differential reductions in terms of the Lax–Sato equations and also in the framework of the dressing method based on the nonlinear Riemann–Hilbert problem.