We perform a detailed analysis of the point transformations which leave invariant the Geng-Xue equation. We find that the Geng-Xue equation admits six Lie point symmetries which possess two three-dimensional subalgebras, the A2,1 ⊗s A1 and A3,8 Lie algebras. For the Lie point symmetries we derive the one-dimensional optimal system and we perform a classification of the corresponding invariant transformations. We demonstrate the application of the Lie symmetries by deriving similarity solutions expressed by closed-form functions.