A pr 2 00 4 On the r - th dispersionless Toda hierarchy I : Factorization problem , symmetries and some solutions

For a family of Poisson algebras, parametrized by r ∈ Z, and an associated Lie algebraic splitting, we consider the factorization of given canonical transformations. In this context we rederive the recently found r-th dispersionless modified KP hierachies and r-th dispersionless Dym hierarchies, giving a new Miura map among them. We also found a new integrable hierarchy which we call the r-th 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 r-th dispersionless KP, r-th dispersionless Dym and r-th dispersionless Toda equations. Some solutions are derived by examining the imposition of a time invariance to the potential r-th 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 r-th dispesionless modified KP equation.