  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 $\mu_k$.
