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We prove the following analogue of the Heyde theorem for a-adic solenoids. Let ξ₁, ξ₂ be independent random variables with values in an a-adic solenoid ${\Sigma }_a$ and with distributions μ₁, μ₂. Let ${\alpha }_j,{\beta }_j$ be topological automorphisms of ${\Sigma }_a$ such that $\beta ₁{\alpha }^{-1}₁\pm \beta ₂{\alpha }^{-1}₂$ are topological automorphisms of ${\Sigma }_a$ too. Assuming that the conditional distribution of the linear form L₂ = β₁ξ₁ + β₂ξ₂ given L₁ = α₁ξ₁ + α₂ξ₂ is symmetric, we describe the possible distributions μ₁, μ₂.