N.M. Stat. § 73-10-15
The secretary of the board of directors must, as soon as the result of any election held under the provisions of this act is declared, enter in the records of such board and file with the county clerk of the county in which the office of said district is located a statement of such results, which statements must show: first, a copy of the published notice of said election; second, the names of the judges of said election; third, the whole number of votes cast in the district and in each precinct of the district; fourth, the names of the persons voted for; fifth, the number of votes given in each precinct for such persons; sixth, the number of votes given in the districts for each of such persons; seventh, the names of the persons declared elected; eighth, the result declared on any question submitted in accordance with the majority of the votes cast for or against such question. The board of directors must declare elected the person having the highest number of votes given for each office, and also the result of any question submitted. The secretary must immediately make out and deliver to such person a certificate of election, signed by him and authenticated with the seal of the board. In case of a vacancy in the board of directors, by death, removal or inability, from any cause, to properly discharge the duties of such director, the vacancy shall be filled by appointment by the remaining members of the board, and upon their failure or inability to act within thirty days after such vacancy occurs, then upon petition of five electors of said district the board of county commissioners of the county where the office of said board of directors is situate, shall fill such vacancy or vacancies. Any director appointed as above provided shall hold his office until the next general election of said district, and until his successor is elected and qualified.
NMS § 73-10-15