Abstract
Binary symmetry constraints are applied to constructing B"acklund
transformations of soliton systems, both continuous and discrete. Construction
of solutions to soliton systems is split into finding solutions to
lower-dimensional Liouville integrable systems, which also paves a way for
separation of variables and exhibits integrability by quadratures for soliton
systems. Illustrative examples are provided for the KdV equation, the AKNS
system of nonlinear Schr"odinger equations, the Toda lattice, and the Langmuir
lattice.
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