  A generalized quintessence model is presented which corresponds to a richer
vacuum structure that, besides a time-dependent, slowly varying scalar field,
contains a varying cosmological term. From first principles we determine a
number of scalar-field potentials that satisfy the constraints imposed by the
field equations and conservations laws, both in the conventional and
generalized quintessence models. Besides inverse-power law solutions, these
potentials are given in terms of hyperbolic functions or the twelve Jacobian
elliptic functions, and are all related to the luminosity distance by means of
an integral equation. Integration of this equation for the different solutions
leads to a large family of cosmological models characterized by luminosity
distance-redshift relations. Out of such models, only four appear to be able to
predict a required accelerating universe conforming to observations on
supernova Ia, at large or moderate redshifts.
