We discuss the origin of two triggers of Swift's Burst Alert Telescope (BAT) that occurred in 2011. The triggers were identified with Swift J185003.2-005627, a previously unknown X-ray source, and the known but unclassified X-ray transient Swift J1922.7-1716. We investigate the BAT data and follow-up observations obtained with Swift's X-ray and ultraviolet/optical telescopes to demonstrate that both triggers are consistent with thermonuclear X-ray bursts. This implies that both sources are neutron star low-mass X-ray binaries. The total duration of 7 minutes and estimated energy output of (3-7) × 1039?erg fall in between that of normal and intermediately long X-ray bursts. From the observed peaks of the X-ray bursts, we estimate a distance of 3.7?kpc for Swift J185003.2-005627 and 4.8?kpc for Swift J1922.7-1716. We characterize the outburst and quiescent X-ray properties of the two sources. They have comparable average outburst luminosities of 1035-36 erg s–1, and a quiescent luminosity equal to or lower than 2 × 1032 erg s–1 (0.5-10?keV). Swift J185003.2-005627 returned to quiescence 20 d after its BAT trigger, while Swift J1922.7-1716 appears to exhibit long accretion outbursts that last several months to years. We identify a unique counterpart for Swift J1922.7-1716 in the ultraviolet/optical data. Finally, we serendipitously detect a flare lasting 500?s from an uncataloged X-ray/optical object that we tentatively classify as a flaring M-dwarf.
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