Finite dimensional Hamiltonian system related to Lax pair with symplectic and cyclic symmetries

For the 1+1 dimensional Lax pair with a symplectic symmetry and cyclic symmetries, it is shown that there is a natural finite dimensional Hamiltonian system related to it by presenting a unified Lax matrix. The Liouville integrability of the derived finite dimensional Hamiltonian systems is proved in a unified way. Any solution of these Hamiltonian systems gives a solution of the original PDE. As an application, the two dimensional hyperbolic $C_n^{(1)}$ Toda equation is considered and the finite dimensional integrable Hamiltonian system related to it is obtained from the general results.