  The ding-a-ling model is a kind of half lattice and half hard-point-gas (HPG)
model. The original ding-a-ling model proposed by Casati {\it et.al} does not
conserve total momentum and has been found to exhibit normal heat conduction
behavior. Recently, a modified ding-a-ling model which conserves total momentum
has been studied and normal heat conduction has also been claimed. In this
work, we propose a full lattice ding-a-ling model without hard point collisions
where total momentum is also conserved. We investigate the heat conduction and
energy diffusion of this full lattice ding-a-ling model with three different
nonlinear inter-particle potential forms. For symmetrical potential lattices,
the thermal conductivities diverges with lattice length and their energy
diffusions are superdiffusive signaturing anomalous heat conduction. For
asymmetrical potential lattices, although the thermal conductivity seems to
converge as the length increases, the energy diffusion is definitely deviating
from normal diffusion behavior indicating anomalous heat conduction as well. No
normal heat conduction behavior can be found for the full lattice ding-a-ling
model.
