Cubature formulas for symmetric measures in higher dimensions with few points

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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 $\ell =5$ or $\ell =7$ 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 $\ell = 2k+1$, that the minimal number of points is almost independent of the weight function. This is also true for the integration over the (Euclidean) sphere.