  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.
