  We investigate the role of the isoscalar vector interaction and the dynamics
of the Polyakov loop on inhomogeneous phases in the phase diagram of the
two-flavor Nambu-Jona--Lasinio (NJL) model. Thereby we concentrate on phases
with a one-dimensional modulation, explicitly domain-wall solitons and, for
comparison, the chiral spiral. While the inclusion of the Polyakov loop merely
leads to quantitative changes compared to the original NJL model, the presence
of a repulsive vector-channel interaction has significant qualitative effects:
Whereas for homogeneous phases the first-order phase transition gets weakened
and eventually turns into a second-order transition or a cross-over, the domain
of inhomogeneous phases is less affected. In particular the location of the
Lifshitz point in terms of temperature and density is not modified.
Consequently, the critical point disappears from the phase diagram and only a
Lifshitz point (showing a different critical behavior) remains. As a result,
susceptibilities remain finite.
