I'd start with an internet search for "XT1 replacement engine." and then make a list of engine models that turn up in your search. A sample result: https://repowerspecialists.com/products/cub-cadet-xt1-engine-replacement?variant=7566434598935
After making a list of engines that supposedly fit, I'd search ebay/amazon/any sites I could find to compare prices on those models.
Youtube may also be a good source for replacement videos that go into the specifics of your model.
Engine mounting bolts need to be the same as does the output shaft diameter and length as your present engine. The throttle linkage should be easy. Whether or not your switch controls will plug directly with your new engine may or may not happen without modification.
I replaced the engine in my JD Z225 last year. It turned out to be a bolt in/plug in easy swap. Since then, it looks like fewer engines are available and prices are up.