Your saddle is a M1936 Philips used strictly by U.S. Cavalry and field artillery officers. Your saddle appears to have all the proper accessories for service in the field: pommel pockets, saddle bags and the cantle shelf.
Philips saddle seats can be found. HOWEVER, it is extremely rare to find a fully equipped Philips. The cantle shelf alone is a $300-$400 collector item. Your complete ensemble is a museum piece. Please do not think of it as a saddle - I urge you to never put it on a horse or attempt to sit on it. No matter how sound it may look, attempting to sit on it will very likely explode the seams of the seat which will basically destroy this historical item.
Think of this ensemble as a piece of antique militaria. Please avoid any synthetic leather treatments. Belvoir glycerine saddle soap is all that is needed and should be applied with just enough water to make the soap bar barely greasy and then gently, not vigorously, rubbed in.
I recommend you not try to unscrew the cantle shelf. It is sheet steel covered in leather. It is flexed to line up the screw flanges with the screw holes. If you take it off, you may not be able to get the shelf to line back up with the holes. Secondly, you may lose one of the screws. They are of a special design and irreplaceable.
The pommel pocks swivel off - no risk in removing them for treatment if you must. Look in the pommel pockets to see if there are two metal bolt heads. They are meant to be screwed into the sockets on the pommel when pockets not in use. If you find the bolt heads, leave them in one of the pockets. They are usually missing from the Philips saddles that are still around. The stirrup irons should be marked "U.S." and dated on the bottom. If the saddle is 100% complete, the stirrup leather buckles should not look like modern ones. They should be slightly offset.