The cubature formulas on a sphere that are invariant with respect to the icosahedral group of rotations

An algorithm for constructing the cubature formulas on a sphere that are invariant with respect to the icosahedral group of rotations is proposed. This algorithm is applied to construct the new cubature formulas that have the algebraic order of accuracy $n=19, 20,21,23,24,25$. Parameters of these cubature formulas are given to 16 significant digits. The table, which contains the main characteristics of all the best today cubature formulas of the icosahedral group of rotations up to the 35th order of accuracy, is given. A real variant of F. Klein's formula that states the connection between the basic invariant forms of the icosahedral group of rotations is given.