N.M. Stat. § 2-10-1
A permanent joint interim committee of the legislature to be called the "legislative education study committee" is created. The committee shall be composed of ten members, four from the senate and six from the house. The house education committee and the senate education committee shall be represented. The committee members shall be appointed for two-year terms which shall expire on the first day of each odd-year session. The term of any member shall terminate when such member ceases to be a member of the legislature. Members shall be appointed by the committees' committee of the senate or, if the appointment is made in the interim, by the president pro tempore after consultation with and agreement of a majority of the members of the committees' committee, and the speaker of the house of representatives; provided, however, minority members shall be appointed by the speaker only from recommendations made by the minority floor leader although the speaker shall retain the right to reject any such recommendations. Vacancies on the committee shall be filled for the unexpired term by the respective appointing authority which makes the original appointments and subject to the same recommendations, provided that members shall be appointed from the respective houses, parties and committees, so as to maintain the same number of house and senate members and the same representation of standing committees as provided in the original appointments. Each of the two parties having the largest number of members in the legislature shall be represented on the committee in proportion to the membership of each such party in each house; provided that in the computation, major fractions shall be counted as whole numbers, and in no event shall either of the two major parties have less than one member from each house.
The officers shall be appointed for terms coterminous with their membership on the committee. Each office shall be alternated between the respective houses each two years. For the terms beginning in 1979, the chairman shall be a house member and the vice chairman, if any, shall be a senate member. The appointing authority of each house shall exercise its appointing authority by naming the chairman or vice chairman respectively on this alternating basis.
No action shall be taken by the committee if a majority of the total membership from either house on the committee rejects such action.
NMS § 2-10-1