  The modified Dirac equations describing massless and massive spin-1/2
particles violating the Lorentz invariance are considered. The equation for
massless fermions with varying speed is formulated in the 16-component
first-order form. The projection matrix, which is the density matrix,
extracting solutions to the equation has been obtained. Exact solutions to the
equation and energy spectrum for a massive neutrino are obtained in the
presence of background matter. We have considered as subluminal as well as
superluminal propagations of neutrinos. The bounds on the Lorentz invariance
violation parameter from astrophysical observations by IceCube, OPERA, MINOS
collaborations and by the SN1987A supernova are obtained for superluminal
neutrinos.
