  We consider a one-dimensional discrete nonlinear Schr{\"o}dinger (dNLS) model
featuring interactions beyond nearest neighbors. We are interested in the
existence (or nonexistence) of phase-shift discrete solitons, which correspond
to four-sites vortex solutions in the standard two-dimensional dNLS model
(square lattice), of which this is a simpler variant. Due to the specific
choice of lengths of the inter-site interactions, the vortex configurations
considered present a degeneracy which causes the standard continuation
techniques to be non-applicable. In the present one-dimensional case, the
existence of a conserved quantity for the soliton profile (the so-called
density current), together with a perturbative construction, leads to the
nonexistence of any phase-shift discrete soliton which is at least $C^2$ with
respect to the small coupling $\epsilon$, in the limit of vanishing $\epsilon$.
If we assume the solution to be only $C^0$ in the same limit of $\epsilon$,
nonexistence is instead proved by studying the bifurcation equation of a
Lyapunov-Schmidt reduction, expanded to suitably high orders. Specifically, we
produce a nonexistence criterion whose efficiency we reveal in the cases of
partial and full degeneracy of approximate solutions obtained via a leading
order expansion.
