  We investigate the charge fluctuations of a large quantum dot coupled to a
two-dimensional electron gas via a quantum point contact following the work of
Matveev. We limit our discussion to the case where exactly two channels enter
the dot and we discuss the role of an anisotropy between the transmission
coefficients (for these two channels) at the constriction. Experimentally, a
channel-anisotropy can be introduced applying a relatively weak in-plane
magnetic field to the system when only one ``orbital'' channel is open. The
magnetic field leads to different transmission amplitudes for spin-up and
spin-down electrons.
  In a strong magnetic field the anisotropic two-channel limit corresponds to
two (spin-polarized) orbital channels entering the dot.
  The physics of the charge fluctuations can be captured using a mapping on the
channel-anisotropic two-channel Kondo model. For the case of weak reflection at
the point contact this has already briefly been stressed by one of us in PRB
{\bf 64}, 161302R (2001). This mapping is also appropriate to discuss the
conductance behavior of a two-contact set-up in strong magnetic field.
  Here, we elaborate on this approach and also discuss an alternative solution
using a mapping on a channel-isotropic Kondo model. In addition we consider the
limit of weak transmission.
  We show that the Coulomb-staircase behavior of the charge in the dot as a
function of the gate voltage, is already smeared out by a small
channel-anisotropy both in the weak- and strong transmission limits.
