My paternal family line were part of that invasion from Normandy. My name is from Normandy and the family stayed in what became England until around 1730 when they ventured to Ameica. Their offspring then joined in the war to secede from the British empire.
My paternal family line has been traced back to the Plantagenets, John Lackland, or King John who sealed Magna Carta, and through succession, to William I. I believe the line of succession was not broken from William the Conqueror to Henry II, making me a descendant of William the Conqueror as well. My ancestor left England for the new world a bit before yours, in 1638. He was a Puritan who was the brother of the rector of St. Mary’s church in Beccles, which is still in existence. That church houses a bust of Lincoln, who’s ancestor migrated with the congregation at the same time.