If his children, &c. Ethan refers to the very words of warning given to Solomon (1Ki_9:6, 1Ki_9:7; Compare Psa_11:11-13), which, with 2Sa 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.