On decompositions of radially symmetric distributions

Let $P_1$ and $P_2$ be probability distributions in $R^n$, $n\ge2$, and $P=P_1*P_2$. If $P$ is radially symmetric (i.e. invariant with respect to rotation around some point) and satisfies the condition
$$ \exists\varepsilon>0\colon P(\{x\in R^n\colon|x|>r\})=O(\exp\{-r^{2+\varepsilon}\}),\quad r\to\infty,\eqno(1) $$
then $P_1$ and $P_2$ must be radially symmetrical too. Condition (1) cannot be weakened by putting $\varepsilon=0$.
A sufficient condition is obtained for a radially symmetric distribution to be indecomposable into two proper distributions. The uniform distribution in the re-dimensional unit ball is shown to be indecomposable for $n\ge3$.