Three-sheeted Riemann surfaces of genus 0 with fixed projections of the branch points

New constructive procedures for determination of the equations for the algebraic functions of the third order and of genus zero are proposed. We discovered that there exist four types of the Riemann surfaces for these functions, which are distinguished by the groups of permutations of the sheets in the branch points. Using the coefficients of the equations we compute the extremal cuts of the Riemann surfaces. These cuts are the attractive sets for the poles of the Hermite–Padé approximants.