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The radio has a constant power feed and is woken up by a bus message when the key is turned on rather than receiving another power feed. If the radio will not shut off most likely that is an internal failure.
The ABS light can be caused by a number of things including wheel speed sensors, the ABS module itself or hydraulic control unit, brake switch, wiring, etc. The most common cause would be a failed speed sensor. The first step in diagnosing this problem will be to use a scan tool to see what fault codes have set and go from there.
The ABS light may be related to the odometer not working. The speedometer and odometer signal originates at the left rear wheel speed sensor, the ABS module picks up the signal and converts it to vehicle speed which is used by the instrument cluster. If the left rear wheel speed signal is missing then it will also take down the speedometer and odometer.
Could this be caused by a bad bump from a pothole, or from a antirust treatment?
thank you for your reply, and an additional detail: the radio works without
the key in the ignition