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

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 $a_i{d}_j-b_i{c}_j\mathrm{\ne }0$ 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.