  Let X be a compact Abelian group. In the article we obtain a characterization
of shifts of Haar distributions on compact open subgroups of the group X by the
symmetry of the conditional distribution of one linear form of independent
random variables taking values in X given another. Coefficients of the linear
forms are topological automorphisms of the group X. This result can be viewed
as an analogue for compact Abelian groups of the well-known Heyde theorem,
where the Gaussian distribution on the real line is characterized by the
symmetry of the conditional distribution of one linear form of independent
random variables given another.
