  We study the singlet-triplet Anderson model (STAM) in which a configuration
with a doublet is hybridized with another with a singlet and a triplet, as a
minimal model to describe two-level quantum dots coupled to two metallic leads
in effectively a one-channel fashion. The model has a quantum phase transition
which separates regions of a doublet and a singlet ground state. The limits of
integer valence of the STAM (which include a model similar to the underscreened
spin-1 Kondo model) are derived and used to predict the behavior of the
conductance through the system at both sides of the transition, where it jumps
abruptly. At a special quantum critical line, the STAM can be mapped to an
ordinary Anderson model (OAM) plus a free spin 1/2. We use this mapping to
obtain the spectral densities of the STAM as a function of those of the OAM at
the transition. Using the non-crossing approximation (NCA), we calculate the
spectral densities and conductance through the system as a function of
temperature and bias voltage, and determine the changes that take place at the
quantum phase transition. The separation of the spectral density into a singlet
and a triplet part allows us to shed light on the underlying physics and to
explain a shoulder observed recently in the zero-bias conductance as a function
of temperature in transport measurements through a single fullerene molecule
[Roch N et al., Nature 453, 633 (2008)]. The structure with three peaks
observed in nonequilibrium transport in these experiments is also explained.
