ver . 2 ON HAMILTONIAN FLOWS ON EULER - TYPE EQUATIONS

Properties of Hamiltonian symmetry flows on hyper-bolic Euler-type Liouvillean equations E ′ EL are analyzed. Description of their Noether symmetries assigned to the integrals for these equations is obtained. The integrals provide Miura transformations from E ′ EL to the multi-component wave equations E. By using these substitutions, we generate an infinite-Hamiltonian commu-tative 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 ⊂ sym E ′ EL is such that these properties are transferred to B and the recursions for E ′ EL are factorized. Two examples associated with the 2D Toda lattice are considered. Introduction. In this paper, we consider the problem of constructing pairs of commutative hierarchies of Hamiltonian evolution equations related by Miura-type transformations and identified with Lie sub-algebras of the Noether symmetry algebras for Euler–Lagrange-type systems. By using two standard schemes ([3, 15, 16]), which are the Miura substitutions defined by the integrals of Liouvillean hyperbolic equations and construction of the second Hamiltonian structure by a Miura transformation, we restrict the exposition to the class of the Euler–Lagrange Liouvillean hyperbolic systems E