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Very-faint X-ray binaries (VFXBs) are a sub-class of black holes and neutronstars in binaries that appear to be accreting at a very low rate. In additionto providing interesting constraints on poorly understood forms of accretion,elucidating the nature of VFXBs is particularly interesting for binaryevolution and population modeling. Through near-infrared (nIR) spectroscopy, wehere investigate the nature of the bursting neutron star and VFXB 1RXHJ173523.7$-$354013 (J1735), which persistently accretes at an X-ray luminosityof $L_X \sim 10^{34} - 10^{35}~L_{\odot}$. Our analysis shows that the nIRemission is dominated by that of the companion star, which we find to be a lateG or early K-type giant, making this the second neutron star identified as aVFXB found to have a giant companion. We discuss how several of the systemproperties are difficult to reconcile with a wind-fed symbiotic X-ray binary.We therefore also propose an alternative scenario wherein J1735 is a widebinary system (supported by the discovery of a 7.5 d modulation in the nIRlight curves) with a quiescent luminosity of $L_X \sim 10^{34} -10^{35}~L_{\odot}$, in which the donor star is overflowing its Roche lobe. Thisraises the possibility that J1735 may, every century or more, exhibit very longand very bright outbursts during which it reaches accretion rates around theEddington limit like the neutron star Z sources.

The nature of very-faint X-ray binaries: Near-infrared spectroscopy of 1RXH J173523.7$-$354013 reveals a giant companion