This sputtering or coughing that your Husqvarna/AYP zero turn radius lawn mower is doing is called dieseling. What is happening is the fuel is still igniting even though there is no spark and the lawn mower is turned off.
The most common cause for dieseling is the lawn mower has stale fuel or has fuel with a lower octane than required. Drain the fuel that is in the tank and replace it with fresh, correct octane fuel.
If the mower continues to sputter then it could be because the wrong spark plug was used, the spark timing was adjusted wrong, or there is carbon buildup in the combustion chamber.
The model and serial numbers on your Husqvarna/AYP zero-turn mower can be found on the frame of the mower.
NOTE: if you need a part for the small engine in your Husqvarna/AYP zero-turn mower, there will be a separate model/serial number for the engine. Here are the small engines that are available in your Husqvarna/AYP mower. Visit the page of the engine you have to find out where the model/serial number is located for your small engine model: