  We consider a particular instance of the truncated realizability problem on
the $d-$dimensional lattice. Namely, given two functions $\rho_1({\bf i})$ and
$\rho_2({\bf i},{\bf j})$ non-negative and symmetric on $\mathbb{Z}^d$, we ask
whether they are the first two correlation functions of a translation invariant
point process. We provide an explicit construction of such a realizing process
for any $d\geq 2$ when the radial distribution has a specific form. We also
derive from this construction a lower bound for the maximal realizable density
and compare it with the already known lower bounds.
