407777 will make that engine an intek V twin
And the manual is called the Intek V-Twin OHV Repair Manual B & S number 273521.
However it covers all of the variations and IMHO the engine repair section of the JD Technical manual for your mower is far better then the B & S manual for the same engine
So my suggestion is to get a copy of the JD manual , ordered directly from JD as a digital download because it only has your exact intek engine in it ( actually it will have a seperate chapter for every engine that was offered .
Down side is the JD price is about 2 hours of a technicians charge out rate, the sort of price that most Americans are too poor to pay.
The B & S manual is all over the web for free .