  A construction of the bi-Hamiltonian structures for integrable systems on
regular time scales is presented. The trace functional on an algebra of
$\delta$-pseudo-differential operators, valid on an arbitrary regular time
scale, is introduced. The linear Poisson tensors and the related Hamiltonians
are derived. The quadratic Poisson tensors is given by the use of the recursion
operators of the Lax hierarchies. The theory is illustrated by
$\Delta$-differential counterparts of Ablowitz-Kaup-Newell-Segur and Kaup-Broer
hierarchies.
