  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 third
observing runs of LIGO and Virgo. The search is enabled by a contemporaneous
timing ephemeris obtained using NICER data. The NICER ephemeris has also been
extended through 2020 October and includes three new glitches. PSR J0537-6910
has the largest spin-down luminosity of any pulsar and is highly active with
regards to glitches. Analyses of its long-term and inter-glitch braking indices
provided intriguing evidence that its spin-down energy budget may include
gravitational-wave emission from a time-varying mass quadrupole moment. Its 62
Hz rotation frequency also puts its possible gravitational-wave emission in the
most sensitive band of LIGO/Virgo detectors. Motivated by these considerations,
we search for gravitational-wave emission at both once and twice the rotation
frequency. We find no signal, however, and report our upper limits. Assuming a
rigidly rotating triaxial star, our constraints reach below the
gravitational-wave spin-down limit for this star for the first time by more
than a factor of two and limit gravitational waves from the $l=m=2$ mode to
account for less than 14% of the spin-down energy budget. The fiducial
equatorial ellipticity is limited to less than about 3e-5, which is the third
best constraint for any young pulsar.
