Anton, I know the manual typically recommends 30wt's, but your 22hp briggs V-twin will do well on any 30 or 40wt variation, synthetic or conventional. I'd personally go with Mobil 1 if want to go synthetic. 5w30 or 10w30 will probably be the easiest to find, but again, any good 30 or 40wt will make the engine outlast the tractor! Nice LT for the money BTW!
G/luck
Joel