This paper is devoted to the study of the underlying linearities of the coupled Harry-Dym （cHD） soliton hierarchy, including the well-known cHD equation. Resorting to the nonlinearization of Lax pairs, a family of finite-dimensional Hamiltonian systems associated with soliton equations are presented, constituting the decomposition of the cHD soliton hierarchy. After suitably introducing the Abel-Jacobi coordinates on a Riemann surface, the cHD soliton hierarchy can be ultimately reduced to linear superpositions, expressed by the Abel-Jacobi variables.
A constrained system associated with a 3 × 3 matrix spectral problem of the nonlinear Schroedinger（NLS） hierarchy is proposed. It is shown that the constrained system is a Hamiltonian system with the rigid body type Poisson structure on the Poisson manifold R^3N. Further, the reduction of the constrained system extended to the common level set of the complex cones is proved to be the constrained AKNS system on C^2N.
