  Commutative rings of one-dimensional difference operators of rank l>1 and
their deformations are effectively constructed. Our analytical constructions
are based on the so-called ''Tyurin parameters'' for the stable framed
holomorphic vector bundles over algebraic curves of the genus equal to g and
Chern number equal to lg. These parameters were heavily used by the present
authors already in 1978-80 for the differential operators. Their deformations
in the discrete case are governed by the 2D Toda Lattice hierarhy instead of
KP. New integrable systems appear here in the case l=2,g=1. The theory of
higher rank difference operators is much more rich than the rank one case where
only 2-point constructions on the spectral curve were used in the previous
literature (i.e. number of 'infinite points'' is equal to 2). One-point
constructions appear in this problem for every even rank l=2k. Only in this
case commutative rings depend on the functional parameters. Two-point
constructions will be studied in the next work: even for higher rank l>1 this
case can be solved in Theta-functions. It is not so for one-point constructions
with rank l>1.
