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 $10^{-17}$.
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