  L. Klebanov proved the following theorem. Let $\xi_1, \dots, \xi_n$ be
independent random variables. Consider linear forms
$L_1=a_1\xi_1+\cdots+a_n\xi_n,$ $L_2=b_1\xi_1+\cdots+b_n\xi_n,$
$L_3=c_1\xi_1+\cdots+c_n\xi_n,$ $L_4=d_1\xi_1+\cdots+d_n\xi_n,$ where the
coefficients $a_j, b_j, c_j, d_j$ are real numbers. If the random vectors
$(L_1,L_2)$ and $(L_3,L_4)$ are identically distributed, then all $\xi_i$ for
which $a_id_j-b_ic_j\neq 0$ for all $j=\overline{1,n}$ are Gaussian random
variables. The present article is devoted to an analog of the Klebanov theorem
in the case when random variables take values in a locally compact Abelian
group and the coefficients of the linear forms are integers.
