Let $ξ_1$, $ξ_2$, $ξ_3$ be independent random variables with nonvanishing characteristic functions, and $a_j$, $b_j$ be real numbers such that $a_i/b_i\ne a_j/b_j$ for $i\ne j$. Let $L_1=a_1ξ_1+a_2ξ_2+a_3ξ_3$, $L_2=b_1ξ_1+b_2ξ_2+b_3ξ_3$. By C.R.Rao's theorem the distribution of the random vector $(L_1, L_2)$ determines the distributions of the random variables $ξ_j$ up to a change of location. We prove an analogue of this theorem for independent random variables with values in a locally compact Abelian group. We also prove an analogue for independent random variables with values in an $a$-adic solenoid of similar C.R.Rao's theorem. In so doing coefficients of linear forms are continuous endomorphisms of the group.