  We prove the non-integrability (non-existence of additional analytic
conserved quantities other than Hamiltonian) for Fermi-Pasta-Ulam (FPU)
lattices by virtue of Lyapunov-Kovalevskaya- -Ziglin-Yoshida's monodromy method
about the variational equations. The key to this analysis is that the normal
variational equations along a certain solution happen to be in a type of Lam\'e
equations. We also introduce the classification problem towards non-homogeneous
nonlinear lattices including FPU lattices using non-integrability preserving
transformation.
