  We demonstrate the existence of special phantom excitations for open and
periodically closed integrable systems at the example of the $XXZ$ Heisenberg
spin chain. The phantom excitations do not contribute to the energy of the
Bethe state and correspond to special solutions to the Bethe Ansatz equations
with infinite "phantom" Bethe roots. The phantom Bethe roots lead to
degeneracies between different magnetization sectors in the periodic case and
to the appearance of spin helix states (SHS), i.e. periodically modulated
states of chiral nature in both open and closed systems. For the periodic
chain, phantom Bethe root (PBR) solutions appear for anisotropies
$\De=\cosh\eta$ with $\exp(\eta)$ being a root of unity, thus restricting the
phenomenon to the critical region $|\De|<1$. For the open chain, PBR solutions
appear for any value of anisotropy, both in the critical and in the
non-critical region, provided that the boundary fields satisfy a criterion
which we derive in this paper. There exist PBR solutions with all Bethe roots
being phantom, and PBR solutions that consist of phantom roots as well as
regular (finite) roots. Implications of our results for an experiment are
discussed.
