Another step forward in searching for cubature formulae with a minimal number of knots for the square

The knots and weights of a cubature formula are determined by a system of nonlinear equations. The number of equations and unknowns can be reduced by imposing some structure on the formula. We are concerned with the construction of cubature formulae which are invariant under rotations. Using invariant theory, we obtain a smaller system of algebraically independent equations. This is used to construct cubature formulae for the square. One of the results is a 24-point formula of degree 11.