  We have investigated analitycally the phase diagram of a generalized
spherical version of the Blume-Emery-Griffiths model that includes
ferromagnetic or antiferromagnetic spin interactions as well as quadrupole
interactions in zero and nonzero magnetic field. We show that in three
dimensions and zero magnetic field a regular paramagnetic-ferromagnetic (PM-FM)
or a paramagnetic-antiferromagnetic (PM-AFM) phase transition occurs whenever
the magnetic spin interactions dominate over the quadrupole interactions.
However, when spin and quadrupole interactions are important, there appears a
reentrant FM-PM or AFM-PM phase transition at low temperatures, in addition to
the regular PM-FM or PM-AFM phase transitions. On the other hand, in a nonzero
homogeneous external magnetic field $H$, we find no evidence of a transition to
the state with spontaneous magnetization for FM interactions in three
dimensions. Nonethelesss, for AFM interactions we do get a scenario similar to
that described above for zero external magnetic field, except that the critical
temperatures are now functions of $H$. We also find two critical field values,
$H_{c1}$, at which the reentrance phenomenon dissapears and $H_{c2}$
($H_{c1}\approx 0.5H_{c2}$), above which the PM-AFM transition temperature
vanishes.
