  The equation of state for ultra-dense matter can be tested from observations
of the ratio of mass to radius of neutron stars. This could be measured
precisely from the redshift of a narrow line produced on the surface. X-rays
bursts have been intensively searched for such features, but so far without
detection. Here instead we search for redshifted lines in the persistent
emission, where the accretion flow dominates over the surface emission. We
discuss the requirements for narrow lines to be produced, and show that narrow
absorption lines from highly ionized iron can potentially be observable in
accreting low mass X-ray binaries (low B field) which have either low spin or
low inclination so that Doppler broadening is small. This selects Serpens X-1
as the only potential candidate persistent LMXB due to its low inclination.
Including surface models in the broad band accretion flow model predicts that
the absorption line from He-like iron at 6.7 keV should be redshifted to $\sim$
5.1 - 5.7 keV (10 - 15 km for $1.4\mathrm{M_\odot}$) and have an equivalent
width of 0.8 - 8.0 eV for surface temperatures of 7 - 10 $\times$ 10$^6$ K. We
use the high resolution Chandra grating data to give a firm upper limit of 2 -
3 eV for an absorption line at $\sim 5$ keV. We discuss possible reasons for
this lack of detection (the surface temperature and the geometry of the
boundary layer etc.). Future instruments with better sensitivity are required
in order to explore the existence of such features.
