  For a discrete, translationally-invariant $\phi^4$ model introduced by
Barashenkov {\it et al.} [Phys. Rev. E {\bf 72}, 35602R (2005)], we provide the
momentum conservation law and demonstrate how the first integral of the static
version of the discrete model can be constructed from a Jacobi elliptic
function (JEF) solution. The first integral can be written in the form of a
nonlinear map from which {\it any} static solution supported by the model can
be constructed. A set of JEF solutions, including the staggered ones, is
derived. We also report on the stability analysis for the static bounded
solutions and exemplify the dynamical behavior of the unstable solutions. This
work provides a road-map, through this illustrative example, on how to fully
analyze translationally-invariant models in terms of their static problem, its
first integral, their full-set of static solutions and associated conservation
laws.
