Decomposition to the modified Jaulent–Miodek hierarchy

From the nonlinearization of Lax pairs (NLP), a family of finite-dimensional Hamiltonian systems (FDHSs) are presented constituting the decomposition of the modified Jaulent–Miodek (mJM) hierarchy. These FDHSs are further proved to be completely integrable in the Liouville sense in view of the generating function method. Finally, the relation between soliton equations and FDHSs is established with the aid of a set of polynomial integrals.