New Mexico Statutes Chapter 3. Municipalities § 3-11-2. Mayor;  vacated office;  appointment by governing body

A. the governing body shall vote at the next meeting immediately following the vacancy to fill the vacancy if the vacancy has not been filled within fifteen days after the vacancy occurred;  and provided further that so long as the vacancy remains unfilled, the item shall be included on each subsequent governing body meeting agenda until the vacancy is filled;

B. the qualified elector appointed to fill the vacancy shall serve until the next regular local election or municipal officer election, whichever is applicable, when a qualified elector shall be elected to fill the remaining unexpired term, if any;  and

C. a resigning mayor shall not select a nominee or be involved in the appointment of a successor to fill a vacancy in the office of mayor.