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We present a search for continuous gravitational-wave signals from the young,energetic X-ray pulsar PSR J0537-6910 using data from the second and thirdobserving runs of LIGO and Virgo. The search is enabled by a contemporaneoustiming ephemeris obtained using NICER data. The NICER ephemeris has also beenextended through 2020 October and includes three new glitches. PSR J0537-6910has the largest spin-down luminosity of any pulsar and is highly active withregards to glitches. Analyses of its long-term and inter-glitch braking indicesprovided intriguing evidence that its spin-down energy budget may includegravitational-wave emission from a time-varying mass quadrupole moment. Its 62Hz rotation frequency also puts its possible gravitational-wave emission in themost sensitive band of LIGO/Virgo detectors. Motivated by these considerations,we search for gravitational-wave emission at both once and twice the rotationfrequency. We find no signal, however, and report our upper limits. Assuming arigidly rotating triaxial star, our constraints reach below thegravitational-wave spin-down limit for this star for the first time by morethan a factor of two and limit gravitational waves from the $l=m=2$ mode toaccount for less than 14% of the spin-down energy budget. The fiducialequatorial ellipticity is limited to less than about 3e-5, which is the thirdbest constraint for any young pulsar.

Diving below the spin-down limit: Constraints on gravitational waves from the energetic young pulsar PSR J0537-6910