Your tractor sounds like it does not have an actual "3 point hitch" feature (in the sense of agriculture implement jargon).
What your industrial tractor has is simply a means of connecting the backhoe to the tractor, not for exchanging a variety of tractor driven accessory implements.
It has been some years since I worked on and around backhoes, but I do remember going to a Ford tractor dealer for parts and when I asked for parts for the old Ford 4500 backhoe, the parts guy asked me if this was a dedicated backhoe or if it was a tractor with a backhoe attachment.
When i asked how to tell the difference, he said that a dedicated backhoe would not have lift arms for implements and that there would be no live PTO shaft present on the rear axle case.
Ours was a dedicated backhoe and I might assume that the one you have is also a dedicated backhoe configuration.
If yours lacks lift arms it probably also lacks a live PTO to the rear.
You could still probably use the tractor to pull a turning plow, but the plow would have to be an "antique" 2 point hitch type of plow that has its own manual lift for the plow built into the plow itself.
I doubt your being able to adapt a 3-point hitch implement to operate on a tractor lacking the necessary lift arms and internal operating cylinder.
I am pretty sure you could custom fit a 3-point conversion package to the tractor (if you could find one), but you would be contemplating a major undertaking that would be both capital and labor intensive.

Yes, it is a dedicated backhoe, without a PTO. It would also be a PITA to remove the backhoe and expensive to make it a workable 3 point hitch.

I have an old CA AC that I have adapted it to before, but it starts spinning the tires if I go more than 6" deep. And I need to break up the hard-pan.

Thanks mownie.

Perhaps you could rig a bar type, single knife, sub soiler to fit the current plow frame and progressively break up the hard pan by criss-crossing the plot repeatedly. Then you might be able to switch the turning plow back to its regular configuration.
You may be able to find a "ready to go", 3-point hitch sub soiler knife implement at a farm supply/equipment dealer and add that to your your AC implement inventory.

Something that just dawned on me while checking local antifreeze prices.

On my AC, I had to get a new rear tire on one and a new tube in the other one a few years back. I forgot to add back the ballast to the tires! DUH!

The AC has the HP and gearing, but not the weight!

Old Timers disease!!!!!!!

Depending on level of need, you can pick up a "subsoiler" at TSC or equivalent for about $150. They also sell a "middle buster" plow for slightly more. I bought the middle buster which has the same frame, hitch, etc. the difference being the wide "furrow out" plow vs a steel shank. Again, the frame, etc. is the same. I fabbed up a subsoiler out of a piece of 3/4" steel by drilling two holes. Used the same mounting bolts as the plow. Now I have both.

Depending on your application, the middle buster may be all you need if it's in a garden or other tilled area. I need the subsoiler to break out and up honeysuckle roots, etc. which keep emigrating from the grown up fence line around the front 1/2 mile of my lawn. Mowing the lawn keeps them from sprouting, but man, can they suck up fertilize! The steel shank rips them up and I just chop them off with an axe. The "furrow" is self sealing.

Good luck,

Ev

Tell me more? Got pictures? Location?