Hamiltonian Flows on Euler-Type Equations

We analyze properties of Hamiltonian symmetry flows on hyperbolic Euler–Liouville-type equations $\mathcal E_{EL}'$. We obtain the description of their Noether symmetries assigned to the integrals of these equations. The integrals provide Miura transformations from $\mathcal E_{EL}'$ to the multicomponent wave equations $\mathcal E$. Using these substitutions, we generate an infinite-Hamiltonian commutative subalgebra $\mathfrak A$ of local Noether symmetry flows on $\mathcal E$ proliferated by weakly nonlocal recursion operators. We demonstrate that the correspondence between the Magri schemes for $\mathfrak A$ and for the induced “modified” Hamiltonian flows $\mathfrak B\subset\operatorname{sym}\mathcal E_{EL}'$ is such that these properties are transferred to $\mathfrak B$ and the recursions for $\mathcal E_{EL}'$ are factored. We consider two examples associated with the two-dimensional Toda lattice.