Heyde theorem on locally compact Abelian groups with the connected component of zero of dimension 1

Let $X$ be a locally compact Abelian group with the connected component of zero of dimension 1. Let $\xi_1$ and $\xi_2$ be independent random variables with values in $X$ with nonvanishing characteristic functions. We prove that if a topological automorphism $\alpha$ of the group $X$ satisfies the condition ${{\rm Ker}(I+\alpha)=\{0\}}$ and the conditional distribution of the linear form ${L_2 = \xi_1 + \alpha\xi_2}$ given ${L_1 = \xi_1 + \xi_2}$ is symmetric, then the distributions of $\xi_j$ are convolutions of Gaussian distributions on $X$ and distributions supported in the subgroup $\{x\in X:2x=0\}$. This result can be viewed as a generalization of the well-known Heyde theorem on the characterization of the Gaussian distribution on the real line.