When a parameterized dynamical system is not (structurally locally) identifiable, it is crucial to reparameterize the system to ensure that the new parameters can be uniquely determined, at least locally. In the article [1], Theorem 2 (also presented in [2] as Theorem 1) claims the existence of an identifiable reparameterization for a parameterized analytic function. We first give a counter-example to show that its conditions are indeed incomplete. Consequently, to address its incompleteness, we will propose a modified version of the theorem.