  Binary Bargmann symmetry constraints are applied to decompose soliton
equations into finite-dimensional Liouville integrable Hamiltonian systems,
generated from so-called constrained flows. The resulting constraints on the
potentials of soliton equations give rise to involutive solutions to soliton
equations, and thus the integrability by quadratures are shown for soliton
equations by the constrained flows. The multi-wave interaction equations
associated with the $3\times 3$ matrix AKNS spectral problem are chosen as an
illustrative example to carry out binary Bargmann symmetry constraints. The Lax
representations and the corresponding ${\bf r}$-matrix formulation are
established for the constrained flows of the multi-wave interaction equations,
and the integrals of motion generated from the Lax representations are utilized
to show the Liouville integrability for the resulting constrained flows.
Finally, involutive solutions to the multi-wave interaction equations are
presented.
