N.Y. Alt. County Gov't Law § 53
A county which shall adopt the county president form of government shall have a county executive to be called "county president." The office of county president shall be elective. The term of office of the county president first elected shall be three years from and including the first day of January of the second calendar year in which the county president form becomes effective in the county. Successors shall be chosen by the electors of the county for terms of four years each. A vacancy occurring otherwise than by expiration of term shall be filled by appointment by the board of supervisors and the person so appointed shall hold office until the commencement of the calendar year next succeeding the first general election at which the vacancy may be filled. The county president may, when authorized by local law, serve as the head of one or more departments not administered by an elective official but without additional compensation. The board of supervisors of the county shall continue to have and exercise all the functions, powers and duties of boards of supervisors as now or hereafter provided by this chapter or by other law, except as limited or otherwise provided by this section. The county president shall have power, as provided in subdivision five of section twenty of the municipal home rule law, to veto any local law passed by the board of supervisors. The foregoing provisions of this section, together with other applicable provisions of this chapter, including those provisions of this chapter which may be made applicable, shall constitute the county president form of government authorized to be adopted by this chapter.
N.Y. Alt. County Gov't Law § 53