  In this paper we study the Klein-Gordon (KG) lattice with periodic boundary
conditions. It is an $N$ degrees of freedom Hamiltonian system with linear
inter-site forces and nonlinear on-site potential, which here is taken to be of
the $\phi^4$ form. First, we prove that the system in consideration is
non-integrable in Liuville sense. The proof is based on the Morales-Ramis
theory. Next, we deal with the resonant Birkhoff normal form of the KG
Hamiltonian, truncated to order four. Due to the choice of potential, the
periodic KG lattice shares the same set of discrete symmetries as the periodic
Fermi-Pasta-Ulam (FPU) chain. Then we show that the above normal form is
integrable. To do this we utilize the results of B. Rink on FPU chains. If $N$
is odd this integrable normal form turns out to be KAM nondegenerate
Hamiltonian. This implies the existence of many invariant tori at low-energy
level in the dynamics of the periodic KG lattice, on which the motion is
quasi-periodic. We also prove that the KG lattice with Dirichlet boundary
conditions (that is, with fixed endpoints) admits an integrable, KAM
nondegenerated normal forth order form, which in turn shows that almost all
low-energetic solutions of KG lattice with fixed endpoints are quasi-periodic.
