If his children, &c. Ethan refers to the very words of warning given to Solomon (1 Kings 9:6; 1 Kings 9:7. Compare Psa 11:11-13), which, with 2 Samuel 7, should be read with this Psalm. Ethan (we may suppose) outlived Solomon, and saw the break-up of the kingdom; and left this Psalm for Instruction (Maschil) for all future time.
children. sons.
And walk not. Figure of speech Pleonasm (App-6), for emphasis.