  Two "elliptic analogues'' of the nonlinear Schr\"odinger hiererchy are
constructed, and their status in the Grassmannian perspective of soliton
equations is elucidated. In addition to the usual fields $u,v$, these elliptic
analogues have new dynamical variables called ``Tyurin parameters,'' which are
connected with a family of vector bundles over the elliptic curve in
consideration. The zero-curvature equations of these systems are formulated by
a sequence of $2 \times 2$ matrices $A_n(z)$, $n = 1,2,...$, of elliptic
functions. In addition to a fixed pole at $z = 0$, these matrices have several
extra poles. Tyurin parameters consist of the coordinates of those poles and
some additional parameters that describe the structure of $A_n(z)$'s. Two
distinct solutions of the auxiliary linear equations are constructed, and shown
to form a Riemann-Hilbert pair with degeneration points. The Riemann-Hilbert
pair is used to define a mapping to an infinite dimensional Grassmann variety.
The elliptic analogues of the nonlinear Schr\"odinger hierarchy are thereby
mapped to a simple dynamical system on a special subset of the Grassmann
variety.
