  We investigate the symmetry component of the center variety of polynomial
differential systems, corresponding to systems with an axis of symmetry in the
real plane. We give a general algorithm to find this irreducible subvariety and
compute its dimension. We show that our methods provide a simple way to compute
the radical of the ideal generated by the focus quantities and, therefore, to
estimate the cyclicity of a center in the case when the ideal is radical. In
particular, we use our methods to get a simple proof of the famous Bautin
theorem on the cyclicity of the quadratic system.
