I remember that Velikovsky made the claim that Israel was actually independent for only a relatively brief time, and that the south fell back under Egyptian suzerainty after the death of Solomon. That, while it had a hebrew king, it also had an Egyptian governor.
At the beginning of his reign, the Israelites came to Rehoboam and asked him to lighten the load which his father had imposed on them. Rehoboam answered them, "I will add to your yoke; my father chastised you with whips, but I will chastise you with scorpions."