This is what you need - Порядок постановки иностранных граждан на учет по месту пребывания и снятие их с учета
Well, by law you should stay at the place of registration, that's the whole purpose of registration
In reality you could stay in one place and be registered in other but that's kinda illegal, however it is very hard for authorities to prove that, so they usually do not bother.