It is common (at least as far as I have seen) for people to sway while praying. However, while saying the first verse of Shema, one musn't do anything else, even walking is forbidden (Shulchan Aruch OC 63:3).

Is it permitted to sway while saying the first verse of Shema, or is it an "activity" which is forbidden?

There are movements that are certainly allowed during the saying of the first verse, so while this source doesn't mention "swaying", the premise that one musn't do "anything else" is not absolute:

Moving the head while reciting the word echad

Some have the custom to move the head while uttering the word echad, as focus is placed on Hashem's mastery over the heavens and the four directions of the world. The head is nodded up, down and in each of the four directions as an indication of this focus. The proper manner to wave the head for this custom is to copy the order of placing the blood of the korbanos on the mizbayach-east, north, west and south. One should avoid waving the head forward and backward and then right and left (or vice versa) so as not to appear to be making the shape of a cross.

(סעיף ו וס"ק כ, עם הרחבת המקור שבשעה"צ ס"ק י)

from Daf HaYomi B'Halacha Daily Email - 19 Kislev/Dec 1, 2015

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