Кубатурные формулы на сфере, инвариантные относительно группы тетраэдра с инверсией

The unified algorithm of searching for the best (in a sense) cubature formulas on a sphere that are invariant under the tetrahedral group of rotations with inversion is described. This algorithm is applied to find parameters of all the best cubature formulas of this symmetry group up to the 41st order of accuracy.