  In this paper, we derive an analytic characterization of the symmetric
extension of a Herglotz-Nevanlinna function in several variables. Here, the
main tools used are the so-called variable non-dependence property and the
symmetry formula satisfied by Herglotz-Nevanlinna function and Cauchy-type
functions. We also provide an extension of the Stieltjes inversion formula for
Cauchy-type functions.
