  We prove the following group analogue of the well-known Heyde theorem on a
characterization of the Gaussian distribution on the real line. Let $X$ be a
second countable locally compact Abelian group containing no subgroups
topologically isomorphic to the 2-dimensional torus. Let $G$ be the subgroup of
$X$ generated by all elements of $X$ of order $2$ and let $\alpha$ be a
topological automorphism of the group $X$ such that ${\rm
Ker}(I+\alpha)=\{0\}$. Let $\xi_1$ and $\xi_2$ be independent random variables
with values in $X$ and distributions $\mu_1$ and $\mu_2$ with nonvanishing
characteristic functions. If the conditional distribution of the linear form
$L_2 = \xi_1 + \alpha\xi_2$ given $L_1 = \xi_1 +\xi_2$ is symmetric, then
$\mu_j$ are convolutions of Gaussian distributions on $X$ and distributions
supported in $G$. We also prove that this theorem is false if $X$ is the
2-dimensional torus.
