  The extended Hubbard model in the zero-bandwidth limit is studied. The
effective Hamiltonian consists of (i) on-site $U$ interaction and intersite
(ii) density-density interaction $W$ and (iii) Ising-like magnetic exchange
interaction $J$ (between the nearest-neighbors). We present rigorous (and
analytical) results obtained within the transfer-matrix method for 1D-chain in
two particular cases: (a) $W=0$ and $n=1$; (b) $U\rightarrow+\infty$ and
$n=1/2$ ($W\neq 0$, $J\neq 0$). We obtain the exact formulas for the partition
functions which enables to calculate thermodynamic properties such as entropy,
specific heat ($c$), and double occupancy per site. In both cases the system
exhibits an interesting temperature dependence of $c$ involving a
characteristic two-peak structure. There are no phase transitions at finite
temperatures and the only transitions occur in the ground state.
