  Binary symmetry constraints of the N-wave interaction equations in 1+1 and
2+1 dimensions are proposed to reduce the N-wave interaction equations into
finite-dimensional Liouville integrable systems. A new involutive and
functionally independent system of polynomial functions is generated from an
arbitrary order square matrix Lax operator and used to show the Liouville
integrability of the constrained flows of the N-wave interaction equations. The
constraints on the potentials resulting from the symmetry constraints give rise
to involutive solutions to the N-wave interaction equations, and thus the
integrability by quadratures are shown for the N-wave interaction equations by
the constrained flows.
