  The linear and nonlinear transport properties of the single-electron
transistor at the degeneracy point are investigated for the case of weak
single-mode tunnel junctions. Two opposing scenarios are considered,
distinguished by whether or not electrons can propagate coherently between the
two tunnel junctions. Each of these two scenarios corresponds to the
realization of a different multichannel Kondo effect - the two-channel Kondo
effect in the case where coherent propagation is allowed, and the four-channel
Kondo effect in the absence of coherent propagation. A detailed analysis of the
linear and nonlinear conductance is presented for each of these scenarios,
within a generalized noncrossing approximation for the nonequilibrium
multichannel Kondo Hamiltonian. A zero-bias anomaly is shown to develop with
decreasing temperature, characterized by the anomalous power laws of the
multichannel Kondo effect. A scaling function of the differential conductance
with V/T (V being the applied voltage bias, T the temperature) is proposed as a
distinctive experimental signature for each of these two scenarios. In the
absence of coherent propagation between the leads, and for asymmetric couplings
to the two leads, a crossover from four-channel to two-channel behavior is
manifested in a vanishing zero-temperature conductance, and in a nonmonotonic
voltage dependence of the differential conductance for small asymmetries.
