First you need to determine the make and model of the transaxle. It may or may not be rebuildable.

I would look for a sheared key between the wheel and axle.

This mower is kept and used away from my residence. I did look at the transaxle. Peerless 2000-002B SilenTrac Series.
Other numbers 2000-002B 1990196.
A sheared key is so easy and cheap to fix that I can only hope that is the problem. I will check that first.

I suspect other manufacturers use this same unit as Murray probably would not have bought a custom unit for their price range equipment. Anyone know of others using the same?

The Peerless transmission used in your Murray (2000-002B) uses a separate hydraulic power pack (794786). The 794786 sells for about $400 new, and a place near me in Princeton has a working used one for sale for $225. Joseph Nemes & Sons.

Not sure whether this is used other than Murray. It shows used on Craftsman DYT-4000 and DLT-3000, which are Husqvarna and AYP/Electrolux. This last bit of info may not be correct. You'll need to research further.

Google results show that the Peerless 2000-002B was used on some Snapper model. Purchased as a Snapper part, the cost is over $1000.00. Some sellers have it for a little over $200.00.
The parts breakdown for the Murray does not show a separate hydraulic power pack nor is such mentioned by those selling the transaxle. I wonder if the hydraulic power pack is a separate unit for powering a lift, front loader, and/or power steering?

I had one of the cheaper model lawn tractors, and it used the type of transmission that you have. It came as a gift, so i found that the splines in the driven pulley, on top of the transmission, had stripped , and was just spinning on the input shaft. A check at several repair ahops turned up a splined pulley that would work, so i bought it for a song, installed it and it worked ok.
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Today with the rear wheels jacked off the ground, turning one wheel by hand caused the other wheel to turn in the opposite direction-- as one expects a normal differential to do. But holding one wheel stationary permitted the other to be turned rather easily. The wheels are tight on the axles and the axles turn when the wheels are rotated. With the rear off the ground, the wheels are turned by the transmission with the engine running. But there is too little power to move the mower. All this could be caused by a sheared key or similar inside the unit. The transaxle is noisy when turned by the belt with the engine running. The driving pulley on the transaxle is held by a nut and seems to be quite solid. A loose driving pulley would not cause the differential problem observed.
Further suggestions or ideas?

shot in the dark - might try the old hydro tranny purge routine - leave the wheels off the ground,run engine full throttle, cycle hydro control full forward to full reverse 5 or 6 times - to get entrapped air bubbles out. Also (don't know this particular tranny), is tranny oil level OK?

Absolutely no sign of leakage from the unit. Not even the dust accumulation that wet oil attracts.

I think you need to go back and read the 2nd reply.

The space between the wheel and axle is quite sufficient to see if the axle and wheel are moving independently from each other and they are not. The wheels are solidly fixed to the axles.

Read carefully here:
WITH THE PARKING BRAKE LOCKED..........and holding ONE Drive tire firmly.........Try to rotate the opposite tire.
If you can still rotate the opposite tire....check the brake disc to see if it is rotating when the drive tire turns. If the disc is turning, that means the brake is not holding well enough to do what we are trying to do here and must be adjusted to lock up before resuming the test.

IF.......you can still turn one drive wheel when the park brake is locked (and the brake disc does not turn) this indicates a stripped spline in the large spur gear and shaft inside the GEARBOX portion of the transaxle (or a broken key in that same shaft-gear match up).
However.....if you are unable to turn a drive wheel as described above..........it narrows the suspect list down to either: Stripped splines on the shaft that connects the HYDRO assembly to the GEARBOX assembly.......or the HYDRO unit itself has suffered internal damage/failure.
But either of the above conditions might render this into a rather cumbersome paperweight as Peerless does not aPeer to offer internal parts for the HYDRO unit and if the shaft from Hydro to gearbox splines are gone, the mating collar in the HYDRO are also gone. But you will have to separate the HYDRO from the GEARBOX to inspect those splines.

If you are unable to turn one wheel as described above with the brake disc locked............the gearbox is probably not defective, but the connecting shaft might still be stripped out inside the HYDRO unit.
If you don't have a trusted helper to assist in watching the brake disc for movement, paint an index mark on it to determine if the brake holds well enough to do the test.

I do appreciate the effort to type the above message. Typing for me is laborious. I have printed out the information and it is now in my folder for that mower. It may be a day or two until I have time to do the indicated tests. Thanks again.

Update and one more question. Searching online found a distributor closing out the particular transaxle for my mower. So I bought one, about $230.00 delivered to my door. Absolutely new in original packaging. At age 70+, installing it was not easy working alone, but it is now installed. The rear wheels removed easily, the keys were in good condition.

I have one small extension spring left over and no obvious place for it. I know it was at the rear on the left side. Could it have been used to insure brake release when the pedal is released?

IF YOU HAVE HAD THE REAR TIRES OFF, MAKE SURE YOU PUT BOTH KEYS BACK IN THE KEY WAYS ON THE AXLE.

bus driver, oh joy! Now you need to look specifically for a small hole on a lever, arm, or bracket where the spring hook might have looped into. If you can find that hole, inspect the wear pattern in the hole (even if a magnifying glass is needed). The side of the hole with obvious wear from the spring points directly toward where the other end hook was anchored. If you can find either hole you may be able to find the other hole more easily.
But if either place the spring hooks to is a tab that is free to spin on a bolt or other pivot.............it will not point where you need to look for the other hook hole.

The press of other matters has kept me from this project since Saturday and will do so for perhaps a few more days. I also have a Craftsman made by AYP that is about 3 years old. Very similar to the Murray. But the seat arrangement on the Murray rides considerably better than the Craftsman. Bumps are really felt on the Sears. The Murray handles uneven terrain with the drive wheels traction better than the Sears even though the Murray is 46" cut and the Sears is 42".
Will post more about the Murray results when they happen.

Completed the transaxle project early this morning. Works fine. The old transaxle problems were internal, but not in the differential portion. The mystery spring does serve to positively release the brake. Thanks to all who offered encouragement.
Going to have to remove the deck again. A strange noise. I know that it can be resolved. I have a MIG welder which sometimes is useful for mower deck work, but I saw nothing needing welding when I cleaned up the deck while it was off the tractor. It is so hot and dry here that the deck problem may be postponed for a few days.

Thanks for keeping your progress updated.

Project is now completed. For those who might have this mower model, the noise problem was the tensioning/idler pulley for the right side blade hitting the pulley for the right side spindle due to a too-long belt. The Murray belt is 037x66. When I replaced the belts last year, my thrifty nature sought lower prices than the genuine Murray belts. Stens (a major aftermarket lawn equipment parts supplier) shows the belt to be 1/2 x 47", their 265-417, so I bought one from some seller on eBay (along with some other stuff). It worked fine until the deck was reinstalled after the transaxle replacement. Today a 1/2 x 46" (A44K, 4L460K) Huskee belt from Tractor Supply was installed and it obviously fits better. This belt drives only one of the 3 blades and is thus loaded much lighter than the main deck drive belt. So I have marked all my records to show that the 46" belt is the better choice for the secondary deck drive belt.

Not intending to clutter the forum with needless posts, the mower now has been used enough to determine that all is working well, the new transaxle and the new belt.

Thumbs up!!