  The general framework for integrable discrete systems on R in particular
containing lattice soliton systems and their q-deformed analogues is presented.
The concept of regular grain structures on R, generated by discrete
one-parameter groups of diffeomorphisms, through which one can define algebras
of shift operators is introduced. Two integrable hierarchies of discrete chains
together with bi-Hamiltonian structures are constructed. Their continuous limit
and the inverse problem based on the deformation quantization scheme are
considered.
