Returning Explorer Puzzle/Variant 2

Puzzle

An explorer walks:

one mile due south,

one mile due east,

and one mile due north.

He finds himself back where he started.

Where are all the places in the world where this is possible?

$(1): \quad$ The North Pole

$(2): \quad$ Any position $1 + \dfrac 1 {2 \pi n}$ miles from the South Pole, where $n$ can be any (strictly) positive integer.

The North Pole solution is explored (no pun intended) in Returning Explorer Puzzle, where he is specifically located, as he then shoots a polar bear.

It remains to explore the region of the South Pole.

Let $S$ denote the South Pole.

Let the explorer starts some distance $x$ from $S$, where $x$ is greater than $1$ mile.

He walks due south to a point $P$ which is $x - 1$ miles from $S$.

He walks $1$ mile due east, in the process circumnavigating $S$ a total of $n$ times, arriving back at $P$.

He then walks due north again, to the place he started from.

Let us treat the location around $S$ as a plane surface.

Thus we have that:

$\ds 2 \pi \paren {x - 1}$

$=$

$\ds \dfrac 1 n$

$\ds \leadsto \ \ $

$\ds x$

$=$

$\ds \dfrac 1 {2 \pi n} + 1$

$\blacksquare$