  Sequences of canonical conservation laws and generalized symmetries for the
lattice Boussinesq and the lattice modified Boussinesq systems are successively
derived. The interpretation of these symmetries as differential-difference
equations leads to corresponding hierarchies of such equations for which
conservation laws and Lax pairs are constructed. Finally, using the continuous
symmetry reduction approach, an integrable, multidimensionally consistent
system of partial differential equations is derived in relation with the
lattice modified Boussinesq system.
