  This paper deals with situations that illustrate how the violation of Lorentz
symmetry in the gauge sector may contribute to magnetic moment generation of
massive neutral particles with spin-1/2 and spin-1. The procedure we adopt here
is based on Relativistic Quantum Mechanics. We work out the non-relativistic
regime that follows from the wave equation corresponding to a certain particle
coupled to an external electromagnetic field and a background that accounts for
the Lorentz symmetry violation, and we read thereby the magnetic dipole moment
operator for the particle under consideration.We keep track of the parameters
that govern the non-minimal electromagnetic coupling and the breaking of
Lorentz symmetry in the expressions we get for the magnetic moments in the
different cases we contemplate. Our claim is that the tiny magnetic dipole
moment of truly elementary neutral particles might signal Lorentz symmetry
violation.
