That appears to be a Honda GCV160 engine; Honda sells a LOT of mower engines to Husky and other mower companies. From the photo, it appears the carburetor is a fixed-throttle, AutoChoke design, as there does not appear to be any cables (throttle / choke) connected to it.
You can get a carb for it (pretty cheap, probably under $25) but first need the
engine serial number off the side of the engine block. It's etched in and could be hard to read, but will be in the form of either 4 or 5 letters followed by a 7-digit number.
With this serial number, you can look up parts online (just google Honda Engine Parts) and you'll find Honda dealers who sell parts online. Parts catalogs will be arranged by the engine serial number. You may also wish to contact your local Honda dealer, who can also help you get the right carb when you give them the engine serial number. Here's a link to find a local dealer, who may actually be cheaper than online, when you figure shipping...FYI...
Find A Honda Dealer