I have a Husqvarna HU675 that has the Kohler xt675 engine and had the same problem. The mower was warrantee for 3 years and took it in for repair just after a year it was purchased. The warrantee was for the mower, not the engine. The engine's warrantee was good for just a year and even though it was only 2 months past they would not honor it.
If you think about it, this is actually a safety hazard and should be recalled. I have seen many posts from different sites having the same issue with this motor.
I believe that over a period of use, the tank opening gets warped from the heat of the engine. It is plastic and directly behind the engine with no firewall between. For mine, I could clearly see the opening was OVAL shaped and therefore had difficulty putting the gas cap on. I ordered a new OEM tank and gas cap. Due to the many attempts to get the cap screwed on right (plastic), the threads eventually got worn. The parts were easily replaced. I added new fuel and the mower ran great after getting it started. Then the next day my wife tried to mow the back but now it starts then immediately shuts off. This time I think the carburetor needs cleaning. This is the last Husqvarna piece of **** I will ever purchase.
First, a chainsaw set my glove on fire because the firewall weld came loose and fell away, then the riding mower didn't have rear locking differential so when I go over uneven lawn and one of the rear tires is off the ground, it just spins and the other rear tire just sits there causing the mower to get stuck. Don't by Husqvarna, the customer support sucks even worse and they don't honor the warrantee or provide shady warrantee. Sorry for the rant. Too much money spent on crap equipment.