  Properties of Hamiltonian symmetry flows on hyperbolic Euler-type Liouvillean
equations E' are analyzed. Description of their Noether symmetries assigned to
the integrals for these equations is obtained. The integrals provide Miura
transformations from E' to the multi-component wave equations E. By using these
substitutions, we generate an infinite-Hamiltonian commutative subalgebra A of
local Noether symmetry flows on E proliferated by weakly nonlocal recursion
operators. We demonstrate that the correlation between the Magri schemes for A
and for the induced "modified" Hamiltonian flows B in the symmetry algebra of
E' is such that these properties are transferred to B and the recursions for E'
are factorized. Two examples associated with the 2D Toda lattice are
considered.
