Independent linear statistics on the two-dimensional torus

Let $X=\mathbf T^2$ be the two-dimensional torus, $\mathrm{Aut}(\mathbf T^2)$ be the group of topological automorphisms of $\mathbf T^2$, $\Gamma(\mathbf T^2)$ be the set of Gaussian distributions on $\mathbf T^2$, and $\xi_1$, $\xi_2$ be independent random variables taking on values in $\mathbf T^2$ and having distributions $\mu_j$ with the nonvanishing characteristic functions. Consider $\delta\in\mathrm{Aut}(\mathbf T^2)$ and assume that the linear forms $L_1=\xi_1+\xi_2$ and $L_2=\xi_1+\delta\xi_2$ are independent. We give the description of possible distributions $\mu_j$ depending on $\delta$. In particular we give the complete description of $\delta$ such that the independence of $L_1$ and $L_2$ implies that $\mu_1,\mu_2\in\Gamma(\mathbf T^2)$. It turned out that the corresponding Gaussian distributions are either degenerate or concentrated on shifts of the same dense in $\mathbf T^2$ one-parameter subgroup.