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Summary form only given. It is well known that electromagnetic waves propagating along the magnetic field direction in the plasma are strongly absorbed when the wave frequency matches the electron cyclotron frequency. This absorption can be eliminated by adding a weak magnetic undulator that leads to the Undulator-Induced Transparency (UIT) of the plasma. Moreover, the group velocity of the waves in the plasma is strongly reduced, resulting in the extreme compression of the wave energy in the plasma. Compressed waves are polarized primarily in the longitudinal direction, and can be used for synchronous electron or ion acceleration. Numerical simulations reveal yet another interesting property of the EM waves in UIT plasma: strong coupling and conversion between two wave polarizations. Depending on how important mode conversion is, several propagation regimes have been identified. We demonstrate that plasma with undulator induced transparency is perfect testing ground for verifying the theory of mode conversion in plasma.