Konopelchenko, BorisMartínez Alonso, LuisMedina, E.2023-06-202023-06-202011-07[1] B. A. Dubrovin and S. P. Novikov, Russian Math. Surveys 44, 35 (1989) [2] D. Y. Benney, Stud. Appl. Math. 52 45-50 (1973). [3] V. E. Zakharov, Func. Anal. Appl. 14, 89 (1980). [4] B. Dubrovin, T. Grava and C. Klein, J. Nonlinear Science 19 57 (2009). [5] C.P. Boyer and J.D. Finley, J. Math. Phys. 23 1126 (1982). [6] M.Mineev-Weinstein, P. Wiegman and A. Zabrodin, Phys. Rev. Lett. 84 5106 (2000). [7] L. Martinez Alonso and E. Medina, J. Phys. A: Math. Gen. 41, 335202 (2008) [8] G. Darboux, Lecons sur la theorie general des surfaces II , Gauthier Villars (1915). [9] V. R. Kudashev and S. E. Sharapov, Phys. Lett. A 154,445 (1991); Theor. Math. Phys. 87, 40 (1991). [10] F. R. Tian, Commun. Pure. Appl. Math. 46 1093 (1993). [11] F. R. Tian, Duke Math. J. 74 203 (1994). [12] B.G. Konopelchenko, L. Martínez Alonso and E. Medina, J.Phys.A:Math.Theor. 43 434020 2010; arXiv:1003.2892. [13] M. V. Pavlov, Hamiltonian formulation of electroforesis equations. Integrable hydrodynamic equations Preprint, Landau Inst. Theor. Phys., Chernogolovka (1987). [14] Y. Kodama and B.G. Konopelchenko, J. Phys. A: Math. Gen. 35, L489- L500 (2002). [15] B.G. Konopelchenko and L. Martínez Alonso, J. Phys. A: Math. Gen. 37, 7859 (2004) [16] I. M. Krichever, Commun. Pure. Appl. Math. 47 437 (1994) [17] L. Alphors, Lectures on quasiconformal mappings, D. van Nostrand Co, Princeton, 1996. [18] V. I. Arnold, Func. Anal. Appl. 6 no.4, 3 (1972) ; Russian Math. Surveys 29 no. 2, 10 (1974); Russian Math. Surveys 30 no. 5, 3 (1975). [19] V. I. Arnold, S. M. Gusein-Zade and A. N. Varchenko, Singularities of differentiable maps, Birkhäuser Boston, Inc. (1985). [20] R. Thom, Structural stability and Morphogenesis: An outline of a General theory of models, Addison-Wesley Reading (1989). [21] B.Konopelchenko, L.Mart´ınez Alonso and E.Medina, On the singular sector of the Hermitian random matrix model in the large N limit, Phys. Lett.A ( to be published), arXiv:1005.4773 (2010). [22] S. P. Tsarev, Sov. Math. Dokl. 31 488 (1985) [23] M.Pavlov, J.Math. Phys. 44 4134 (2003).0040-577910.1007/s11232-011-0078-yhttps://hdl.handle.net/20.500.14352/44790©Springer. This work was supported by the Spanish Ministerio de Educación y Ciencia (Research Project No. FIS2008-00200/FIS).We completely describe the singular sectors of the one-layer Benney system (classical long-wave equation) and dispersionless Toda system. The associated Euler-Poisson-Darboux equations E(1/2, 1/2) and E(-1/2,-1/2) are the main tool in the analysis. We give a complete list of solutions of the one-layer Benney system depending on two parameters and belonging to the singular sector. We discuss the relation between Euler-Poisson-Darboux equations E(ɛ, ɛ) with the opposite sign of ɛ.engjournal articlehttp://dx.doi.org/10.1007/s11232-011-0078-yhttp://link.springer.comopen access51-73Nonlinear schrodinger-equationHydrodynamic symmetriesWhitham equationsDeformationsHierarchyCurvesFísica-Modelos matemáticosFísica matemática