Summary: Let \( {D^2} \subset {\mathbb{R}^2} \) be a closed unit 2-disk centered at the origin \( O \in {\mathbb{R}^2} \) and let \(F\) be a smooth vector field such that \(O\) is a unique singular point of \(F\) and all other orbits of \(F\) are simple closed curves wrapping once around \(O\). Thus, topologically \(O\) is a “center” singularity. Let \( \theta :D2\backslash \left\{ O \right\} \to \left( {0, + \infty} \right) \) be the function associating with each \(z \neq O\) its period with respect to \(F\). In general, such a function cannot be even continuously defined at \(O\). Let also \( {\mathcal{D}^{+}}(F) \) be the group of diffeomorphisms of \(D^{2}\) that preserve orientation and leave invariant each orbit of \(F\). It is proved that \(\theta\) smoothly extends to all of \(D^{2}\) if and only if the 1-jet of \(F\) at \(O\) is a “rotation,” i.e., \( {j^1}F(O) = - y\frac{\partial}{{\partial x}} + x\frac{\partial}{{\partial y}} \). Then \( {\mathcal{D}^{+}}(F) \) is homotopy equivalent to a circle.