Independent random variables on Abelian groups with independent sum and difference

Let $X$ be a second countable locally compact Abelian group. Let $\xi_1$, $\xi_2$ be independent random variables with values in the group $X$ and distributions $\mu_1$, $\mu_2$ such that the sum $\xi_1+\xi_2$ and the difference $\xi_1-\xi_2$ are independent. Assuming that the connected component of the zero of group $X$ contains a finite number of elements of order 2, we describe the possible distributions $\mu_k$.