  The second-order modified Dirac equation leading to the modified dispersion
relation due to the Lorentz invariance violation corrections is suggested. The
equation is formulated in the 16-component first- order form. I have obtained
the projection matrix extracting solutions of the equation with definite spin
projections which can be considered as the density matrix for pure spin states.
Exact solutions of the equation are found for particles in the external
constant and uniform magnetic field. The synchrotron radiation radius within
the novel modified Dirac equation is estimated.
