Cubature formulas for symmetric measures in higher dimensions with few points

We study cubature formulas for d-dimensional integrals with an arbitrary symmetric weight function of product form. We present a construction that yields a high polynomial exactness: for fixed degree = 5 or = 7 and large dimension d the number of knots is only slightly larger than the lower bound of Möller and much smaller compared to the known constructions. We also show, for any odd degree = 2k + 1, that the minimal number of points is almost independent of the weight function. This is also true for the integration over the (Euclidean) sphere.