Sputter, cough, hiccup

[duckpin]

My 105 has had this slight issue lately where after running it after 10-15 minutes with the deck engaged, the engine will start to cough and sputter and nearly stall. I'll disengage the blades and take the throttle down so it catches itself, and then slowly move the throttle back up to high and engage the deck. Sometimes I won't have an issue after that, sometimes I'll have to continue doing that for another 5-10 minutes. But the bottom line is this only happens after the mower has been running for a little while and has been mowing the grass. Any ideas? I figured if it was the carb or the fuel filter, the issue would arise at the beginning and continue throughout, or at least not run well after it arises.

But at least I got it to mow the lawn today for the first time in a month ... tried like hell to put on a new mule drive belt that just would not fit, despite it being the correct part number. I find out (after bringing the tractor to the shop this morning) that the previous owner did something to lengthen the area where the deck attaches to the handle, thus making the correct belt not work. Fortunately, a slightly longer one did the trick.