  Let D be a closed unit 2-disk on the plane centered at the origin 0, and F be
a smooth vector field on D 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 p:D\0-->(0,+\infty) be the function associating to each $z\not=O$ its
period with respect to F. This function can be discontinuous at O.
  Let Diff(F) be the group of all diffeomorphisms of D which preserve
orientation and orbits of F. We prove that p smoothly extends to all of D if
and only if the 1-jet of F at the origin is a non-degenerate linear map, and
that in this case Diff(F) is homotopy equivalent to the circle.
