We describe directed searches for continuous gravitational waves (GWs) from 16 well-localized candidate neutron stars, assuming none of the stars has a binary companion. The searches were directed toward 15 supernova remnants and Fomalhaut b, a directly imaged extrasolar planet candidate that has been suggested to be a nearby old neutron star. Each search covered a broad band of frequencies and first and second time derivatives. After coherently integrating spans of data from the first Advanced LIGO observing run of 3.5-53.7. days per search, applying data-based vetoes, and discounting known instrumental artifacts, we found no astrophysical signals. We set upper limits on intrinsic GW strain as strict as 1 x 10(-25), fiducial neutron star ellipticity as strict as 2 x 10(-9), and fiducial r-mode amplitude as strict as 3 x 10(-8).
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