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©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., Dirección General de Enseñanza Superior e Investigación Científica, Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©IOP Publishing. The authors are grateful to Prof. Manuel Mañas for many useful discussions., Solutions of the Riemann-Hilbert problem implementing the twistorial structure of the dispersionless Toda (dToda) hierarchy are obtained. Two types of string equations are considered which characterize solutions arising in hodograph sectors and integrable structures of two-dimensional quantum gravity and Laplacian growth problems., Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©IOP Publishing. The authors wish to thank the Spanish Ministerio de Ciencia e Innovación, research projects FIS2005-00319 and FIS2008-00200, and acknowledge the support received from the European Science Foundation (ESF) and the activity entitled Methods of Integrable Systems, Geometry, Applied Mathematics (MISGAM). This paper was finished during the research visits of one of the authors (MM) to the Université Catholique de Louvain and to the Scuola Internazionale Superiore di Studi Avanzati/International School for Advanced Studies (SISSA) in Trieste, MM wish to thanks Prof. van Moerbeke and Prof. Dubrovin for their warm hospitality, acknowledge economical support from MISGAM and SISSA and reckons different conversations with P. van Moerbeke, T. Grava, G. Carlet and M. Caffasso., The multicomponent 2D Toda hierarchy is analyzed through a factorization problem associated with an infinite-dimensional group. A new set of discrete flows is considered and the corresponding Lax and Zakharov-Shabat equations are characterized. Reductions of block Toeplitz and Hankel bi-infinite matrix types are proposed and studied. Orlov-Schulman operators, string equations and additional symmetries (discrete and continuous) are considered. The continuous-discrete Lax equations are shown to be equivalent to a factorization problem as well as to a set of string equations. A congruence method to derive site-independent equations is presented and used to derive equations in the discrete multicomponent KP sector (and also for its modification) of the theory as well as dispersive Whitham equations., Ministerio de Ciencia e Innovación (MICINN), European Science Foundation (ESF), MISGAM, SISSA, Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

© 2003 IOP Publishing Ltd. This work originated during the stay of the authors at the Isaac Newton Institute for the Mathematical Sciences of Cambridge University as participants of the programme ‘Integrable Systems’. The authors are grateful to the organizers for the support provided. They also acknowledge S P Tsarev and A Mikhailov for useful comments and conversations. The work is partially supported by CICYT proyecto PB98–0821., A general scheme for analysing reductions of Whitham hierarchies is presented. It is based on a method for determining the S-function by means of a system of first-order partial differential equations. Compatibility systems of differential equations characterizing both reductions and hodograph solutions of Whitham hierarchies are obtained. The method is illustrated by exhibiting solutions of integrable models such as the dispersionless Toda equation (heavenly equation) and the generalized Benney system., CICYT, Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©IOP Publishing. The author thanks Luis Martínez Alonso and Elena Medina for several discussions. Partial economical support from Dirección General de Enseñanza Superior e Investigación Científica no. BFM2002-01607 is also acknowledged., For a family of Poisson algebras, parametrized by r is an element of Z, and an associated Lie algebraic splitting, we consider the factorization of given canonical transformations. In this context, we rederive the recently found rth dispersionless modified KP hierarchies and rth dispersionless Dym hierarchies, giving a new Miura map among them. We also found a new integrable hierarchy which we call the rth dispersionless Toda hierarchy. Moreover, additional symmetries for these hierarchies are studied in detail and new symmetries depending on arbitrary functions are explicitly constructed for the rth dispersionless KP, rth dispersionless Dym and rth dispersionless Toda equations. Some solutions are derived by examining the imposition of a time invariance to the potential rth dispersionless Dym equation, for which a complete integral is presented and, therefore, an appropriate envelope leads to a general solution. Symmetries and Miura maps are applied to get new solutions and solutions of the rth dispersionless modified KP equation., Dirección General de Enseñanza Superior e Investigación Científica, Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©2003 IOP Publishing Ltd. This work is partially supported by the DGESIC Proyecto PB98-0821., The factorization problem for the group of canonical transformations close to the identity and the corresponding twistor equations for an ample family of canonical variables are considered. A method to deal with these reductions is developed for the construction of classes of nontrivial solutions of the dKP equation., DGESIC, Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub