If the city previously was organized under the council form of government, the first president and the other members of the first board of city commissioners elected after a city has incorporated under the city commission system of government are the successors of the mayor and council members of the city. Upon the qualification of the president and other members of the board, all the powers, rights, and duties of the mayor and council of the city cease. Whenever the city previously, under the decree or judgment of any court or under any law, ordinance, or resolution, has been entitled to representation through the mayor of the city and one or more of the council members of the city, on the board of directors of any incorporated company in which the city may own stock or be interested, it must be represented on the board of directors by the president of the board of city commissioners and by two other members of the board to be selected by the board.