I've checked the homepage of the BrainsBreaker and this is what it says when it comes to the license retrieval which I believe it will apply to your case as well:
If you bought the program's license in the last five years, or purchased any additional set of puzzles in this time, we will resend your activation code. Please email email@example.com and tell us your name, the email and home addresses you had at the time you ordered and also the approximate date of your purchase and we will email you your activation code.
As you can see, you have to email the developers to the specified email address so they can issue a new code to register the application on your second PC. Also, the previous game needs to be uninstalled.