Singular sectors of the one-layer Benney and dispersionless Toda systems and their interrelations

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.