  For a family of Poisson algebras, parametrized by by an integer number r, 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.
