  We study geometric and algebraic geometric properties of the continuous and
discrete Neumann systems on cotangent bundles of Stiefel varieties $V_{n,r}$.
The systems are integrable in the non-commutative sense, and by applying a
$2r\times 2r$--Lax representation, we show that generic complex invariant
manifolds are open subsets of affine (non-compact) Prym varieties on which the
complex flow is linear. The characteristics of the varieties and the direction
of the flow are calculated explicitly. Next, we construct a family of
(multi-valued) integrable discretizations of the Neumann systems and describe
them as translations on the Prym varieties, which are written explicitly in
terms of divisors of points on the spectral curve. It appears that the systems
inherit or naturally generalize the basic properties of the classical Neumann
system on $S^{n-1}$ and, therefore, of the Jacobi--Mumford systems: the
structure of the Lax matrices, the spectral curve, the equations of motion,
linearization on Abelian varieties, and, in the discrete case, the formula for
the translation on them.
