Squaring the Circle and Cubing the Sphere: Circular and Spherical Copulas

Do there exist circular and spherical copulas in $R^d$? That is, do there exist circularly symmetric distributions on the unit disk in $R^2$ and spherically symmetric distributions on the unit ball in $R^d$, $d\ge3$, whose one-dimensional marginal distributions are uniform? The answer is yes for $d=2$ and 3, where the circular and spherical copulas are unique and can be determined explicitly, but no for $d\ge4$. A one-parameter family of elliptical bivariate copulas is obtained from the unique circular copula in $R^2$ by oblique coordinate transformations. Copulas obtained by a non-linear transformation of a uniform distribution on the unit ball in $R^d$ are also described, and determined explicitly for $d=2$.