N.M. Stat. § 10-6-2
The officer, agent, employee, board or other agency of the state of New Mexico, or of its departments, agencies, counties, municipalities or political subdivisions, who is by law authorized to fill ordinary vacancies in the public office or employment so temporarily abandoned as provided in Section 1 [10-6-1 NMSA 1978] hereof is hereby authorized, empowered and directed to appoint to the public office or employment, so abandoned as provided in Section 1 hereof, another person who shall receive the salary and perform the duties thereof until, but only until, the former incumbent shall have been relieved from active duty in the armed services and shall have resumed the personal discharge of the duties of such public office or employment.
NMS § 10-6-2