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 tensor product form. We present a construction that yields a high polynomial exactness: for fixed degree l=5 or l=7 and large dimension, the number of knots is only slightly larger than the lower bound of M"oller and much smaller compared to the known constructions. We also show, for any odd degree l=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.