Inthispaper,anewcompletelyintegrablesystemrelatedtothecomplexspectralproblem—φ_(xx)+(i/4)uφ_x+(i/4)(uφ)_x+(1/4)vφ=iλφ_xandtheconstrainedBowsoftheBoussinesqequationsaregenerated.AccordingtotheviewpointofHamiltonianmechanics,theEuler-LagrangeequationsandtheLegendretransformations,areasonableJacobi-Ostrogradskycoordinatesystemisobtained.Moreover,bymeansoftheconstrainedconditionsbetweenthepotentialu,vandtheeigenfunctionφ,theinvolutiverepresentationsofthesolutionsfortheBoussinesqequationhierarchyaregiven.
