(Abridged) Results from the first dedicated study of Galactic PNe by means of optical integral field spectroscopy with the VLT FLAMES Argus IFU are presented. Three typical Galactic-disk PNe have been mapped with the 11.5x7.2 Argus array: two dimensional spectral maps of NGC 5882, 6153 and 7009 with 297 spatial pixels per target were obtained at sub-arcsec resolutions and 297 spectra per target were obtained in the 396.4-507.8 nm range. Spatially resolved maps of emission lines and of nebular physical properties were produced. The abundances of helium and of doubly ionized carbon and oxygen were derived from optical recombination lines (ORLs), while those of O^2+ were also derived from the collisionally excited lines (CELs). The abundance discrepancy problem was investigated by mapping the ratio of ORL/CEL abundances for O^2+ (the abundance discrepancy factor; ADF) across the face of the PNe. The ADF varies between targets and also with position within the targets attaining values of ~40 in the case of NGC 6153. Correlations of the ADF with geometric distance from the nucleus, as well as with [O III] electron temperature, plasma ionization state and other physical properties are established. Very small values of the temperature fluctuation parameter in the plane of the sky are found in all cases. It is argued that these results provide further evidence for the existence in typical PNe of a distinct nebular component consisting of hydrogen-deficient plasma. The zones containing this posited component appear as undulations in the C II and C II ORL abundance diagnostics of about 2 spatial pixels across; any associated structures should have physical sizes of less than ~1000 astronomical units. We propose that circumstellar disks, Abell 30-type knots, or Helix-type cometary globules may be involved.
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