  We give the proof of the real analyticity of the accessory parameters in
Liouville field theory as a function of the position of the sources in the case
in which in addition to elliptic sources, parabolic sources are present. The
method is a non trivial extension of the elliptic case as it requires in an
intermediate step the introduction of a regulator. The treatment holds also in
the case of the torus. A discussion is given of the extension to higher genus
surfaces.
