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We show that the $m$-dimensional Euler--Manakov top on $so^*(m)$ can berepresented as a Poisson reduction of an integrable Hamiltonian system on asymplectic extended Stiefel variety $\bar{\cal V}(k,m)$, and present its Laxrepresentation with a rational parameter.  We also describe an integrable two-valued symplectic map $\cal B$ on the4-dimensional variety ${\cal V}(2,3)$. 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 classicalEuler top in space and has a transparent geometric interpretation, which can beregarded as a discrete version of the celebrated Poinsot model of motion andwhich inherits some properties of another discrete system, the ellipticbilliard.  The reduction of $\cal B$ to $so^*(3)$ gives a new explicit discretization ofthe Euler top in the angular momentum space, which preserves first integrals ofthe continuous system.Comment: 18 pages, 1 Figur

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