  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 $\bar{\cal V}(k,m)$, and present its Lax
representation with a rational parameter.
  We also describe an integrable two-valued symplectic map $\cal B$ on the
4-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 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 $\cal B$ 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.
