The prophets promise that a new David will arise to reign over a united kingdom. Judah and Israel will be bound back together under a single king.
That sounds like a prophetic extension of David’s story, but in fact it’s something more like a simple repetition. David was first anointed king, after all, during Saul’s reign, and from that point to Saul’s death the kingdom of Israel was divided between the small band of warriors around David and the company of king Saul.
The new David unites a divided Israel – just like the original David.