Characterization of probability distributions on some locally compact Abelian groups containing an element of order 2

The well-known Heyde theorem characterizes the Gaussian distributions on the real line by the symmetry of the conditional distribution of one linear form of independent random variables given another. We generalize this theorem to groups of the form $\mathbb{R}\times F$, where $F$ is a finite Abelian group such that its 2-component is isomorphic to the additive group of the integers modulo $2$. In so doing, coefficients of the linear forms are arbitrary topological automorphisms of the group. Previously, a similar result was proved in the case when the group $F$ contains no elements of order 2. The presence of an element of order 2 in $F$ leads to the fact that a new class of probability distributions is characterized