That hoe is a 33 series backhoe that fits a case 480F backhoe, it has a single boom cylinder. If it has two boom cylinders it is a 35 series and would work on a 580 series up to the E series. Like said above case made a heavy duty 3 point hitch that you could remove the hitch and install the backhoe, it took a good 30+ min to do this with the correct tools on hand and the correct tube kit to hook the backhoe hoses to. The later 580K was a non removable main frame and a option of a bolt on if the customer had plans to buy a 3 point frame and be able to switch it. The 570 size machine would handle it just not on a 3 point, due to it needing to be more ridget. You would need to look at your parts book and do some welding to make up a adapter plate and plan for the hyd hook ups. There is a lot of 3 point backhoe's on the market both new and used. Some of the issues with them are the seating, climbing back to it, the backhoe attachment moves around on the links due to it not being ridget when placing down pressure on the boom, and not being able to use the tractor weight to really make it stable while digging. You may get lucky and find a dealer that has the correct backhoe just sitting on his equipment line and wants to move it. i would research your parts book for that machine and see how the backhoe mounts from the plant. If that hoe shown could be bought cheap i would try it just to get the depth and reach for digging and more lifting power. Just my take.
