  We present a directed search for continuous gravitational wave (CW) signals
emitted by spinning neutron stars located in the inner parsecs of the Galactic
Center (GC). Compelling evidence for the presence of a numerous population of
neutron stars has been reported in the literature, turning this region into a
very interesting place to look for CWs. In this search, data from the full O3
LIGO--Virgo run in the detector frequency band $[10,2000]\rm~Hz$ have been
used. No significant detection was found and 95$\%$ confidence level upper
limits on the signal strain amplitude were computed, over the full search band,
with the deepest limit of about $7.6\times 10^{-26}$ at $\simeq 142\rm~Hz$.
These results are significantly more constraining than those reported in
previous searches. We use these limits to put constraints on the fiducial
neutron star ellipticity and r-mode amplitude. These limits can be also
translated into constraints in the black hole mass -- boson mass plane for a
hypothetical population of boson clouds around spinning black holes located in
the GC.
