We study the singlet.triplet Anderson model (STAM) in which a configuration with a doublet is hybridized with another containing 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 on both sides of the transition, where it jumps abruptly. At a special quantum critical line, the STAM can be mapped to an infinite-U 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 et al, Nature 453 633 (2008)]. The structure with three peaks observed in nonequilibrium transport in these experiments is also explained.