Hello Crumpet, prd and aussietrev
I would love to identify this one - because of the use of a
Greenfield-style drive.

I think the extra pulley could be the input from the PTO side of the engine.
I think one can see the chain drive sprocket to the rear axle.
This means that the other serpentine belt also powers the cutter deck.
This is just a guess.

Cheers
----------------
Jack

p.s. I am not keen on the above-ground swimming pool seat.