In this paper we give a necessary condition for structural identifiability of uncontrolled autonomous systems. This condition only turns on the identifiability of the right-hand side of the nonlinear differential system. We prove that this necessary condition becomes sufficient when the state is one-dimensional, To the best of our knowledge, the theoretical results obtained in this paper are new although the proofs are trivial. But they give an easy to check condition as it is shown by the study of some typical examples, in which we do much less computation than the involved litterature to prove identifiability properties.