Witnesses told a number of Calgary media outlets say the man shouted "white power" when he appeared at the rally in the downtown on Saturday.
Police say in a news release that officers followed the man, but he fled.
They say the officers caught up to him but there was a physical struggle when one officer approached him.
They say the man produced a metal pipe, and that within seconds, a police firearm was discharged.
The man was taken to hospital where he remains in critical condition, while the officer suffered injuries and was treated at the scene.
The Alberta Serious Incident Response Team, a provincial agency that investigates police, has been called in to investigate the shooting.
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