  High-precision tests of local Lorentz invariance, via monitoring of the
sidereal time variation of the photon energies emitted by ultrarelativistic
heavy-ion beams and of the beam momentum, are proposed. This paper includes
descriptions of the physics ideas and the concept for the detector. The
experiment results will allow high-precision tests of LLI via anisotropy of the
maximum attainable speed of a photon and an ion. The projected accuracy for the
asymmetries interpreted in the framework of the anisotropic relativistic
mechanics corresponds to the limit on sidereal time variation of the one-way
maximum attainable speed at the levels between $10^{-14}$ and $10^{-17}$.
