My vague recollection is there are no movement / parking restrictions but some flights need to be notified.
I know of a British guy who keeps a 'G' reg. Bulldog out in the US.
The Bulldog never had an FAA type certificate, so it couldn't be put on the 'N' reg.
There are no restrictions on keeping foreign reg. aircraft in the States. I DO know that you ARE required to notify authorities of every movement that a 'foreign registered' biz jet makes out there.