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Commun. Theor. Phys. (Beijing, China) 52 (2009) pp. 1081–1086 c Chinese Physical Society and IOP Publishing Ltd Vol. 52, No. 6, December 15, 2009 A Finite-Dimensional Completely Integrable System Associated with Boussinesq Hierarchy CHEN Lan-Xin1,2,∗ and ZHANG Jun-Xian2 1China University of Mining and Technology, Beijing 100083, China 2Shijiazhuang University, Shijiazhuang 050035, China (Received December 31, 2008) Abstract In this paper, a new completely integrable system related to the complex spectral problem −ϕxx +(i/4)uϕx + (i/4)(uϕ)x + (1/4)vϕ = iλϕx and the constrained flows of the Boussinesq equations are generated. According to the viewpoint of Hamiltonian mechanics, the Euler–Lagrange equations and the Legendre transformations, a reasonable Jacobi–Ostrogradsky coordinate system is obtained. Moreover, by means of the constrained conditions between the potential u, v and the eigenfunction ϕ, the involutive representations of the solutions for the Boussinesq equation hierarchy are given. PACS numbers: 45.20.Jj Key words: eigenvalue problem, constraint flow, symplectic manifold, completely integrability, involutive rep- resentation 1 Introduction equations, a new finite-dimensional completely integrable In 1975, Kaup[1] derived a nonlinear evolution equa- system is given. Furthermore, the representations of the tion from a higher-order water-wave equation which is the solution for the Boussinesq hiererchy are obtained by the following classical Boussinesq equation commutable flows of the finite-dimensional classical com- 1