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Not without building a whole rectifier and voltage limiting circuit. The rear LED needs a 6V DC current. The dynamo produces an AC output with much higher voltages - around 60V I believe.
P. S. The fact the front light is still working suggests it's just a loose connection and that the circuits inside the front light are still working. If there's any soldering to do, I suspect its either inside the front light or in the wiring between the two.
So my supernova pure front light failed and is not sending any power to rear light, but it still shines well. So was wondering is there any way to connect the rear supernova light directly to the dynamo hub? maybe soldering a resistor or something inline (not a electrician, but can solder basic stuff)