Аннотация:
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 Bäcklund 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.