Elec. Code, § 10515

(a) If, by 5 p.m. on the 83rd day prior to the day fixed for the general district election: (1) only one person has filed a declaration of candidacy for any elective office to be filled at that election, (2) no one has filed a declaration of candidacy for such an office, (3) in the case of directors to be elected from the district at large, the number of persons who have filed a declaration of candidacy for director at large does not exceed the number of offices of director at large to be filled at that election, or (4) in the case of directors who must reside in a division but be elected at large, the number of candidates for director at large from a division does not exceed the number required to be elected director at large while residing in that division; and if a petition signed by 10 percent of the voters or 50 voters, whichever is the smaller number, in the district or division if elected by division, requesting that the general district election be held has not been presented to the officer conducting the election, he or she shall submit a certificate of these facts to the supervising authority and request that the supervising authority, at a regular or special meeting held prior to the Monday before the first Friday in December in which the election is held, appoint to the office or offices the person or persons, if any, who have filed declarations of candidacy. The supervising authority shall make these appointments.

(b) If no person has filed a declaration of candidacy for any office, the supervising authority shall appoint any person to the office who is qualified on the date when the election would have been held. The person appointed shall qualify and take office and serve exactly as if elected at a general district election for the office.

(c) Where a director must be appointed to represent a division, all or most of which is not within the county governed by the supervising authority, then the board of supervisors of the county within which all or most of that division is located shall be the body to which request for appointment is made and which shall make the appointment.