Let $X$ be a locally compact Abelian group, $α_{j}, β_j$ be topological automorphisms of $X$. Let $ξ_1, ξ_2$ be independent random variables with values in $X$ and distributions $μ_j$ with non-vanishing characteristic functions. It is known that if $X$ contains no subgroup topologically isomorphic to the circle group $\mathbb{T}$, then the independence of the linear forms $L_1=α_1ξ_1+α_2ξ_2$ and $L_2=β_1ξ_1+β_2ξ_2$ implies that $μ_j$ are Gaussian distributions. We prove that if $X$ contains no subgroup topologically isomorphic to $\mathbb{T}^2$, then the independence of $L_1$ and $L_2$ implies that $μ_j$ are either Gaussian distributions or convolutions of Gaussian distributions and signed measures supported in a subgroup of $X$ generated by an element of order 2. The proof is based on solving the Skitovich-Darmois functional equation on some locally compact Abelian groups.