  New classes of generalized Nevanlinna functions, which under multiplication
with an arbitrary fixed symmetric rational function remain generalized
Nevanlinna functions, are introduced. Characterizations for these classes of
functions are established by connecting the canonical factorizations of the
product function and the original generalized Nevanlinna function in a
constructive manner. Also a detailed functional analytic treatment of these
classes of functions is carried out by investigating the connection between the
realizations of the product function and the original function. The operator
theoretic treatment of these realizations is based on the notions of rigged
spaces, boundary triplets, and associated Weyl functions.
