(a) If the mayor or commissioner of a municipality dies or resigns, the other members of the governing body of the municipality shall appoint a person to fill the vacancy. A person serving as a member of the governing body is not, because of that service, ineligible to be appointed to fill a vacancy in the office of mayor of the municipality, but the person may not vote on the person’s own appointment.
(b) If, because of death, resignation, failure to qualify, or other reason, vacancies exist in the offices of mayor and commissioner at the same time or in the offices of two commissioners at the same time, the county judge shall order a special election to fill the vacancies. The election is governed by the provisions applicable to an election under Subchapter A, Chapter 8.

(c) The county judge shall certify the results of the election to the clerk of the governing body and the clerk shall enter the results in the minutes.