SDCL § 9-13-14.3: Elected official to hold office until vacancy filled.

If for any reason a municipality fails to elect any person to succeed an elected official whose term has expired or an elected official fails to file a nominating petition or qualify, the office is deemed vacant. The elected official whose term has expired shall continue to act in an official capacity until the vacancy is filled by election or appointment pursuant to § 9-13-14.1 or 9-13-14.2.