The Klebanov theorem for the group $\mathbb{R}\times \mathbb{Z}(2)$

L. Klebanov proved the following theorem. Let $ξ_1, \dots, ξ_n$ be independent random variables. Consider linear forms $L_1=a_1ξ_1+\cdots+a_nξ_n,$ $L_2=b_1ξ_1+\cdots+b_nξ_n,$ $L_3=c_1ξ_1+\cdots+c_nξ_n,$ $L_4=d_1ξ_1+\cdots+d_nξ_n,$ 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 $ξ_i$ 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 analogue of the Klebanov theorem in the case when random variables take values in the group $\mathbb{R}\times \mathbb{Z}(2)$ and the coefficients of the linear forms are topological endomorphisms of this group.