Cubature formulas on the sphere that are invariant under the transformations of the dihedral group of rotations $D_4$

An algorithm for finding the best cubature formulas (in a sense) on the sphere that are invariant under the transformations of the dihedral group of rotations $D_4$ is described. This algorithm is applied for finding parameters of all the best cubature formulas of this symmetry type up to the 35th order of accuracy.