Stationary Flows Revisited

In this paper we revisit the subject of stationary flows Lax hierarchies a coupled KdV class. We explain main ideas in standard case and then consider dispersive water waves (DWW) case, with respectively 2 3 Hamiltonian representations. Each representation gives us different form flow. Comparing these, construct Poisson maps, which, being non-canonical, give rise to bi-Hamiltonian representations flows. An alternative approach is use Miura which do DWW hierarchy, has two ''modifications''. This structure sequences related maps build tri-Hamiltonian each three hierarchies. One allows multi-component squared eigenfunction expansion, N degrees freedom Hamiltonians, first integrals. A for derived from matrices. freedom, generalisation our matrices functions, connection rational Calogero-Moser (CM) system. coupling CM system other potentials, along representation. present particular one integrable Hénon-Heiles systems CM.