Abstract: By using the constraint relating potential and eigenfunctions, the decomposition of each equation in the Boussinesq hierarchy into two commuting finite-dimensional integrable Hamiltonian system (FDIHS) is presented. A method to construct the Lax representations for both x-and t_n-constrained flows via reduction of the adjoint representations of the auxiliary linear problems is developed.

Key words: constrained flow, Lax representation, decomposition, adjoint representation, zero-curvature equation