A vehicle for machining a rail by grinding and/or planing, including a metal abrasive module, an eccentric drive performing an eccentric movement, at least two abrasive blocks connected with the eccentric drive, and a force exerting drive pressing at least one abrasive block onto the rail, wherein the metal abrasive module and/or at least one of the abrasive blocks is individually tiltable around an axis parallel to the rail, and wherein at least one of the abrasive blocks is individually pressable onto the rail. This allows a more effective and improved metal removal and rail shaping capabilities.