Suggest you check out a few of the you-tube video's about engine-swapping, if you've never done it before, you can make the job fairly easy,..
(using same make and APPROX same HP rating,) most important is diameter & length of the crankshaft (so you can re-use your current "stack-pulley" and/or electric PTO clutch assembly) mounting bolt holes remain the same, wiring MAY have minor differences, if any, muffler has to be thought of as well,then your engine throttle and/or choke cable(s) that's about it.
Staying with the same make, but going from say a single to a twin, think everything mentioned previously, but now engine size MIGHT come into play, will it fit under your mower's HOOD ..?? usually, NOT a problem, BUT be aware it's not always the case, might encounter muffler problems, ( new brackets, plates etc.) fuel-tank location may enter into the picture now, as I stated watch a few of the vids and get an idea of where you're headed and what's involved, also take LOTS of pics, BEFORE, DURING & some AFTER, they help more than you can imagine, especially with wiring and carburetor linkages/ controls.
Changing from one make to another, although done quite often, it involves some headaches, usually the wiring, the stack-pulley/PTO clutch, mounting holes, muffler, etc. as I said, watch the videos and then determine which route you're going to take, repair as Bert pointed out, which can be done, or replace, whichever you decide, best of luck, and keep us up-dated. ..:thumbsup:
https://www.youtube.com/watch?v=_grHmGj3TS8 ........ one of several videos