“A bear in the window” problem, revisited

Some months ago, I posted a nice problem:

I walked 10 Km to the South, then I turned left, walking straight 10 Km to the East, and then I turned again, going 10 Km to the North, directly arriving to my home. Then I opened a window, and I saw a bear in the outside. Where my house is?

The easy solution was the North Pole.

The not easy solution was… there is some Earth parallel (like the red one of the picture) with a perimeter of just 10 Km, near the South Pole (actually, 89º59′49″ S), thereby my house could be 10 Km to the North of of that parallel. In this way, I walk 10 Km to the South (arriving to the magic parallel), then I walk 10 Km to the East (doing just a full turn), and finally 10 Km to the North, returning to the same initial place.

The funny thing is that we (my flatmate and I) have discovered (by mistake) another solution…
There are more places on Earth’s surface with the same property!! A lot more!! Try to find out.

You can find the answer in the first comment of this post. A clue: Harmonics.