  This work presents the first integral field spectroscopy of the Homunculus
nebula around Eta Carinae in the near-infrared spectral region (J band). We
confirmed the presence of a hole on the polar region of each lobe, as indicated
by previous near-IR long-slit spectra and mid-IR images. The holes can be
described as a cylinder of height (i.e. the thickness of the lobe) and diameter
of 6.5 and 6.0x10^{16} cm, respectively. We also mapped the blue-shifted
component of He I 10830 seen towards the NW lobe. Contrary to previous works,
we suggested that this blue-shifted component is not related to the Paddle but
it is indeed in the equatorial disc.
  We confirmed the claim of Smith (2005) and showed that the spatial extent of
the Little Homunculus matches remarkably well the radio continuum emission at 3
cm, indicating that the Little Homunculus can be regarded as a small HII
region. Therefore, we used the optically-thin 1.3 mm radio flux to derive a
lower limit for the number of Lyman-continuum photons of the central source in
Eta Car. In the context of a binary system, and assuming that the ionising flux
comes entirely from the hot companion star, the lower limit for its spectral
type and luminosity class ranges from O5.5 III to O7 I. Moreover, we showed
that the radio peak at 1.7 arcsec NW from the central star is in the same
line-of-sight of the `Sr-filament' but they are obviously spatially separated,
while the blue-shifted component of He I 10830 may be related to the radio peak
and can be explained by the ultraviolet radiation from the companion star.
