  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.
