Just use plain sign.
I signed the book.
Sign on means something different - it means to register availability somehow, to be available over a communications medium, or the equivalent to logging in, i.e. sign on to that account.
If you say
I signed on to the official register of ...
it makes me think you just logged into a website.
Now, sign on with or signed on for might be interpreted as registered with, subscribed to, or joined by signing up for, but usually would be sign up with or sign up for.
I signed on with the official register of ... (Sounds OK)
I signed up with the official register of ... (Sounds better)
But none of those mean physically, actually signing something.