  Let $X$ be a compact connected Abelian group. It is well-known that then
there exist topological automorphisms $\alpha_j, \beta_j $ of $X$ and
independent random variables $\xi_1$ and $\xi_2$ with values in $X$ and
distributions $\mu_1, \mu_2$ such that the linear forms $L_1 = \alpha_1\xi_1 +
\alpha_2\xi_2$ and $L_2 = \beta_1\xi_1 + \beta_2\xi_2$ are independent, whereas
$\mu_1$ and $\mu_2$ are not represented as convolutions of Gaussian and
idempotent distributions. This means that the Skitovich--Darmois theorem fails
for such groups. We prove that if we consider three linear forms of three
independent random variables taking values in $X$, where $X$ is an
${\boldsymbol a}$-adic solenoid, then the independence of the linear forms
implies that at least one of the distributions is idempotent. We describe all
such solenoids.
