When creating a new request using Live support, the current user is set as owner automatically.
The value is actually set as soon as you click on "Live Support"/Rendering and we are not able to change this behavior.
It seems like this value is set in a code not accessible from smax gui.
Many requests created in Live Support are not resolved at first contacts. Then best practice is to assign the request to an assignment group, not a person. And customers have to manually remove the value in the owner field before saving the request.
We needs this functionality to be a part of rules accessible for us to change. Then customers can choose if they want this value automatically set or not.