Integrable flows and Bäcklund transformations on extended Stiefel varieties with application to the Euler top on the Lie group SO(3)

We show that the m-dimensional Euler–Manakov top on so∗(m) can be represented as a Poisson reduction of an integrable Hamiltonian system on a symplectic extended Stiefel variety V̄(k,m), and present its Lax representation with a rational parameter. We also describe an integrable two-valued symplectic map B on the 4dimensional variety V(2, 3). The map admits two different reductions, namely, to the Lie group SO(3) and to the coalgebra so∗(3). The first reduction provides a discretization of the motion of the classical Euler top in space and has a transparent geometric interpretation, which can be regarded as a discrete version of the celebrated Poinsot model of motion and which inherits some properties of another discrete system, the elliptic billiard. The reduction of B to so∗(3) gives a new explicit discretization of the Euler top in the angular momentum space, which preserves first integrals of the continuous system.