  We investigate the charge fluctuations of a single-electron box (metallic
grain) coupled to a lead via a smaller quantum dot in the Kondo regime. The
most interesting aspect of this problem resides in the interplay between spin
Kondo physics stemming from the screening of the spin of the small dot and
orbital Kondo physics emerging when charging states of the grain with (charge)
Q=0 and Q=e are almost degenerate. Combining Wilson's numerical
renormalization-group method with perturbative scaling approaches we push
forward our previous work [K. Le Hur and P. Simon, Phys. Rev. B 67, 201308R
(2003)]. We emphasize that for symmetric and slightly asymmetric barriers, the
strong entanglement of charge and spin flip events in this setup inevitably
results in a non trivial stable SU(4) Kondo fixed point near the degeneracy
points of the grain. By analogy with a small dot sandwiched between two leads,
the ground state is Fermi-liquid like which considerably smears out the Coulomb
staircase behavior and hampers the Matveev logarithmic singularity to arise.
Most notably, the associated Kondo temperature $T_K^{SU(4)}$ might be raised
compared to that in the conductance experiments through a small quantum dot
$(\sim 1K)$ which makes the observation of our predictions a priori accessible.
We discuss the robustness of the SU(4) correlated state against the inclusion
of an external magnetic field, a deviation from the degeneracy points,
particle-hole symmetry in the small dot, asymmetric tunnel junctions and
comment on the different crossovers.
