  Tiny violation of Lorentz invariance has been the subject of theoretic study
and experimental test for a long time. We use the Standard-Model Extension
(SME) framework to investigate the effect of the minimal Lorentz violation on
the structure of a neutron star. A set of hydrostatic equations with
modifications from Lorentz violation are derived, and then the modifications
are isolated and added to the Tolman-Oppenheimer-Volkoff (TOV) equation as the
leading-order Lorentz-violation corrections in relativistic systems. A
perturbation solution to the leading-order modified TOV equations is found. The
quadrupole moments due to the anisotropy in the structure of neutron stars are
calculated and used to estimate the quadrupole radiation of a spinning neutron
star with the same deformation. The calculation puts forward a new test for
Lorentz invariance in the strong-field regime when continuous gravitational
waves are observed in the future.
