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 $1.4mathrm{M_odot}$) 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.
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