On a decomposition of a Gaussian distribution on groups

Let $X$ be a connected locally compact Abelian separable metric group.
The following generalization of Cramer's theorem is obtained: an arbitrary Gaussian distribution $\mu$ on the group $X$ has only Gaussian divisors if and only if $X$ does not contain a subgroup isomorphic to the circle group T.
It is also shown that any Gaussian distribution $\mu$, the support of which coincides with $X$, has a non-Gaussian divisor if and only if the group $X$ is isomorphic to a group of the form $R^p\times T$, $p\ge 0$.