  The Hamiltonian representation for the hierarchy of Lax-type flows on a dual
space to the Lie algebra of shift operators coupled with suitable
eigenfunctions and adjoint eigenfunctions evolutions of associated spectral
problems is found by means of a specially constructed Backlund transformation.
The Hamiltonian description for the corresponding set of squared eigenfunction
symmetry hierarchies is represented. The relation of these hierarchies with Lax
integrable (2+1)-dimensional differential-difference systems and their triple
Lax-type linearizations is analysed. The existence problem of a Hamiltonian
representation for the coupled Lax-type hierarchy on a dual space to the
central extension of the shift operator Lie algebra is solved also.
