Mañas Baena, Manuel Enrique2023-06-202023-06-202004-11-120305-447010.1088/0305-4470/37/46/007https://hdl.handle.net/20.500.14352/51534©IOP Publishing. The author is in debt with Luis Martínez Alonso and Elena Medina, coauthors of [18], paper on which the present one is partially based. Partial economical support from Dirección General de Enseñanza Superior e Investigación Científica no BFM2002-01607 is also acknowledge.We introduce an S-function formulation for the recently found rth dispersionless modified KP and rth dispersionless Dym hierarchies, giving also a connection of these S-functions with the Orlov functions of the hierarchies. Then, we discuss a reduction scheme for the hierarchies that together with the S-function formulation leads to hodograph systems for the associated solutions. We consider also the connection of these reductions with those of the dispersionless KP hierarchy and with hydrodynamic-type systems. In particular, for the one-component and two-component reduction we derive, for both hierarchies, ample sets of examples of explicit solutions.engjournal articlehttp://dx.doi.org/10.1088/0305-4470/37/46/007http://iopscience.iop.orghttp://arxiv.org/abs/nlin/0405028open access51-73Topological field-theoryR-Matrix theoryIntegrable hierarchiesWhitham hierarchyAdditional symmetriesConformal-mapsDkp equationTau-functionSystemsDeformationsFísica-Modelos matemáticosFísica matemática