The conditions are that you must sell your existing property first.
If you buy a new property with a mortgage from BoI, I very much doubt that Ulster would allow you to move your tracker to replace the BoI mortgage.
You have a few options to explore
1) Don't go ahead with the purchase of the new home until you have sold your existing home
2) Ask Ulster Bank if they will give you a mortgage on the new home pending the sale of your existing home. They are far more likely to go for that than to refinance a BoI mortgage
3) If you have exchanged contracts on the new house, ask a family member to buy it instead. Then when you sell your house you can buy the house from your family member. It will cost you 1% stamp duty and legal fees, but may well be worth it to keep the tracker.
Brendan