Nic,
A picture is worth 1,000 words in this case.
I didn't understand your dilemma until I checked Parts Lookup, where no fewer than fourteen (14) different models with 4-digit numbers are listed; each configured with two cylinder motors (not counting serial number variants, or ones that I may have overlooked, see listing below).
If the motor has not been changed, most of them were configured with Kohler motors, either a KT17 or an M18; models 1641, 1863, and 1864 models all used a motor with a V-Twin configuration.
This leaves only Model 1604 which came with a 16 HP Briggs & Statton (like a 582) as a possible candidate. Does your motor look like the B&S depicted below?
[ATTACH=full]18145[/ATTACH]
The one I own looks like this:
[ATTACH=full]18146[/ATTACH]
(Note the flywheel is not installed [ATTACH=full]18147[/ATTACH], by the way, the parts of this motor still usable are for sale.)
Of the Models I checked those with Kohler Opposed Twin included:
Model 1535
Model 1541
Model 1710
Model 1711
Model 1806
Model 1810
Model 1811
Model 1860
Model 1861
Model 1862
Other engine/model combinations mentioned above.
Hope this helps. [ATTACH=full]18148[/ATTACH]
Edit: Oops, I forgot you don't have the motor. Oh well.