keep a low profile

keep a low profile (to avoid attracting attention to yourself; to behave in a way that does not attract attention ) — держаться в тени; залечь на дно; затихариться (разг.)

Example 1: He's been keeping a low profile at work ever since his argument with Pete. (CIDI)

Example 2: he local people are not always pleasant to strangers, who are warned to keep a low profile while visiting. (Cambridge Dictionary of American Idioms)

see also
[lay low for a while]
[make oneself scarce]
[on the QT]
[under the table]

keep out of the public eye

Example : Carolyn keeps out of the public eye and away from Washington. (Cambridge Dictionary of American Idioms)

Сf.
[play possum]