  For a five-parameter discrete $\phi^4$ model, we derive various exact static
solutions, including the staggered ones, in the form of the basic Jacobi
elliptic functions $\sn$, $\cn$, and $\dn$, and also in the form of their
hyperbolic function limits such as kink ($\tanh$) and single-humped pulse
($\sech$) solutions. Such solutions are admitted by the considered model in
seven cases, two of which have been discussed in the literature, and the
remaining five cases are addressed here. We also obtain $\sin$e, staggered
$\sin$e as well as a large number of short-periodic static solutions of the
generalized 5-parameter model. All the Jacobi elliptic, hyperbolic and
trigonometric function solutions (including the staggered ones) are
translationally invariant (TI), i.e., they can be shifted along the lattice by
an arbitrary $x_0$, but among the short-periodic solutions there are both TI
and non-TI solutions. The stability of these solutions is also investigated.
Finally, the constructed Jacobi elliptic function solutions reveal four new
types of cubic nonlinearity with the TI property.
