King Beli of Sogn (a traditional district in Western Norway) had two sons and a daughter named Ingeborg. Helgi was his first son, and Halfdan his second. On the other side of the fjord, lived the king's friend Thorstein (Þorsteinn Víkingsson) whose son Frithjof (Friðþjófr) was called the bold (hinn frœkni). Frithiof was the tallest, strongest and he was the bravest among men.
When the king's children were but young their mother died. A goodman of Sogn named Hilding (Hildingr), prayed to have the king's daughter to foster. Frithjof was the foster-brother to the king's daughter as he was also raised together with Ingeborg (Ingibjörg) by their foster-father Hilding.
Both Beli and Þorsteinn died in war whereupon Helgi and Halfdan took over the kingdom. The two kings were jealous of Frithjof's excellent qualities and so they denied him Ingeborg's hand. They took her to Baldr's sacred enclosure Baldrshagi where no one dared hurt another and where no woman and man had intercourse. Still Frithjof visited Ingeborg and they continued to love each other. This caused Helgi and Halfdan to send Frithjof away to Orkney to take tribute and while he was away they burnt down his homestead and married Ingeborg to King Ring, the aged king of Ringerike.
When Frithjof returned with the tribute, he burnt down Baldr's temple in Baldrshagi and went away to live as a viking. After three years, he came to King Ring and spent the winter with him. Just before the old king died, Frithjof's identity was apparent to everybody and so the dying king appointed Frithjof earl and made him the care-taker of Ring's and Ingeborg's child.
When Ring had died, Frithjof and Ingeborg married and he became the king of Ringerike. Then he declared war on Ingeborg's brothers, killed one of them and made the second one his vassal.

Question: Who took over running the kingdom after the death of the King?
Answer: Helgi and Halfdan
Alternative answer: the King's sons, Helgi and Halfdan