We got a phone call from a family member about an hour ago saying they found this great fare on the web. The company is EuroFly USA and we just booked our air from JFK to Bologna, one way, for about $1,600.00 for 2 adults and 2 kids.
This is the best deal we have seen by far. EuroFly seems to be a reputable carrier - we will let you know!
Anthony