you should just be able to copy over the license file from the old machine to the new.
i think how this works is you get an auth code, and that generates a license file, and then the auth code is invalid from that point on. however, the license file can be copied to whatever computers you are using, so you don't need to go thru the reauthorization step. of course that implies that you have a license file that was generated with 1.8.6 or later as shown in dave's link.
rob