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.

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