Despite a harsh decree to divide the kingdom due to Solomon's sins, a remnant of the monarchy is preserved for the Davidic dynasty. God's original choice of this royal line remains intact, though Solomon's actions dictate that his descendants will no longer rule over the majority of the nation [ביאור שטיינזלץ].
Although the promise mentions leaving a single tribe under the rule of the House of David, commentators agree that this actually refers to two tribes: Judah and Benjamin. They are treated as one because they share a combined territory. Since the royal city of Jerusalem sits squarely within the land of both tribes, Benjamin was naturally absorbed into and became secondary to the much larger, central tribe of Judah [רד״ק, רלב״ג, מצודת דוד].
This partial preservation of the kingdom stems from two main reasons, the first being the merit of David. Divine protection granted to a father extends to his children. It is because of David that the kingdom is not divided during Solomon's own lifetime, and it is through his merit that his descendants do not lose the throne entirely [רלב״ג]. This grace is not granted because of any merit of Solomon himself, but solely because David was a devoted servant of God who received a promise that his throne would endure forever [מלבי״ם, אברבנאל].
The second reason is the holy status of Jerusalem. God's choice of Jerusalem as the eternal home for the Temple requires its ruler to be a direct descendant of David and Solomon, the very builders of the city and the sanctuary. A king who is a direct descendant of the Temple's builders will naturally be more dedicated and diligent in maintaining it, acting out of deep respect for the legacy of his ancestors [מצודת דוד, רלב״ג, אברבנאל]. The very selection of Zion as a holy site guarantees that the throne of David will permanently remain there [מלבי״ם].
A broader historical perspective emerges when comparing this account to the Book of Chronicles. The Book of Kings does not shy away from criticism, openly detailing Solomon's sins, his foreign wives, and the severe prophecy regarding the division of his kingdom. In contrast, Ezra the Scribe intentionally omitted these harsh details when writing Chronicles. His purpose in that book was to focus exclusively on Solomon's praises rather than his failures, leading him to leave out the events that brought about the punishment of a divided kingdom [אברבנאל].