  The spin-1 Ising-Heisenberg diamond chain with the second-neighbor
interaction between the nodal spins is rigorously solved using the
transfer-matrix method. Exact results for the ground state, magnetization
process and specific heat are presented and discussed in particular. It is
shown that the further-neighbor interaction between the nodal spins gives rise
to three novel ground states with a translationally broken symmetry, but at the
same time, it does not increases the total number of intermediate plateaus in a
zero-temperature magnetization curve compared with the simplified model without
this interaction term. The zero-field specific heat displays interesting
thermal dependencies with a single- or double-peak structure.
