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 Schrodinger(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 R3N. 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 C2N.