https://www.youtube.com/watch?v=pnuLTeHB3yU

There is only one day/night cycle at the South Pole with Polar noon in 3 weeks on the Solstice. The residual daily rotation at the South Pole still captures daily noon even though technically rotational speeds reduce to zero from a maximum speed of 1037.5 miles per hour at the Equator.

Like everywhere else on the planet, the Polar sundial will register the changes in the time for each 360 degree revolution of the shadow around the gnomon (flag pole) reflecting two surface rotations, one daily where the normal experience of daylight/darkness is absent and the other orbital where there is just one daylight/darkness cycle each year.