(a) In cities or incorporated towns not having heretofore adopted a home rule charter pursuant to Chapter 8 of this title, the board of adjustment shall consist of all of the following members or their authorized agents:
(1) The chief engineer of the street and sewer department, the public works commissioner, or the city manager.
(2) The city solicitor.
(3) The mayor.
If the city or incorporated town has no city engineer or public works commissioner, or city solicitor, then the mayor or chief executive of such city or town shall appoint 2 members, each to be appointed for a term of 3 years and removable for cause by the appointing authority upon written charges and after public hearing, who, with the presiding officer of the zoning commission, shall constitute the board of adjustment for such city or town. Vacancies shall be filled for the unexpired term of any member whose term becomes vacant.
(b) In cities or incorporated towns having heretofore or hereafter adopted a home rule charter pursuant to Chapter 8 of this title, the legislative body thereof may establish a board of adjustment consisting of 5 members who shall be residents of the city or incorporated town and who shall have knowledge of and experience in the problems of urban or suburban or rural development, and who, at the time of appointment, shall not be candidates-elect for or incumbents of an elective public office. The mayor or chief executive officer of such city or incorporated town, with consent of the legislative body thereof, shall appoint 4 members for terms of 4 years, provided that the terms of the original members shall be established in a manner that 1 shall expire each year. The mayor or chief executive officer of such city or incorporated town, with the consent of the legislative body thereof, shall appoint 1 member who shall be chairperson and who shall serve at the pleasure of that appointing official. The members shall be entitled to compensation as determined by the city or incorporated town.
(c) In the event that a city or incorporated town qualifying under subsection (b) of this section fails to establish a board of adjustment as permitted in subsection (b) of this section, the board of adjustment shall consist of those persons designated in subsection (a) of this section.
(d) (1) Anything heretofore in this section to the contrary notwithstanding, any city or town, by its legislative body, may establish a board of adjustment consisting of not less than 3 nor more than 5 members who shall be residents of the city or town and who shall have knowledge of the problems of urban or suburban or rural development and who, at the time of appointment and throughout the term of office, shall not be candidates nor members of the legislative body nor employees of the city or town. The mayor or chief executive officer of such city or town shall appoint such members of the board of adjustment, and all such appointments shall be confirmed by a majority vote of the elected members of the legislative body.
(2) All appointments shall be for a period of 3 years, provided that the terms of the original members shall be established in such a manner that the term of at least 1 member shall expire each year and the successor shall be appointed for a term of 3 years. The board of adjustment so selected shall elect from among their own number a chairperson and a secretary.
(3) Any member of the board of adjustment may be removed from office by the legislative body for cause after a hearing by a majority vote of all the elected members of the legislative body of such city or town. A vacancy occurring otherwise than by the expiration of term shall be filled for the remainder of the unexpired term in the same manner as an original appointment.