Two newton decompositions of staionary flows of KdV and Harry Dym hierarchies

We show that each stationary flow of the KdV and Harry Dym hierarchies of soliton equations, which are (2m+1)-st order ODEs (m=0,1…), has two parametrisations as a set of Newton equations with velocity-independent forces. Forces are potential and these Newton equations follow from a Lagrangian function with an inefinite kinetic energy term. These two parametrisations are canonically inequivalent and give rise to new bihamiltonian structures in classical mechanics. Lax representations for these Newton equations are found.