  The wave equation for spinless particles with the Lorentz violating term is
considered. We formulate the third-order in derivatives wave equation leading
to the modified dispersion relation. The first-order formalism is considered
and the density matrix is obtained. The Schr\"{o}dinger form of equations is
presented and the quantum-mechanical Hamiltonian is found. Exact solutions of
the wave equation are obtained for particles in the constant and uniform
external magnetic field. The change of the synchrotron radiation radius due to
quantum gravity corrections is calculated.
