Independent random variables on Abelian groups with independent the 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 zero of the group $X$ contains a finite number elements of order 2 we describe the possible distributions ${\mu }_k$.