The mermaid statue is by icelandic sculptor Nína Sæmundsson. It is a recast from the original mold, as the original 1948 statue was damaged many years ago. The statue was originally placed in the lake, then was later moved to a couple different locations in the park near city hall, and was eventually reinstalled in the lake as you see here.
As in many other cultures, there is a myth about a mermaid in Iceland who sits on a rock in the sea and attracts sailors with her song - after which they disappear forever into the depths in her embrace.
● The Mermaids of Earth coffee-table book: See page 56 in the book about this sculpture.