  We propose to use a model-independent criterion based on first integrals of
motion, due to Noether symmetries of the equations of motion, in order to
classify the dark energy models in the context of scalar field (quintessence or
phantom) FLRW cosmologies. In general, the Noether symmetries play an important
role in physics because they can be used to simplify a given system of
differential equations as well as to determine the integrability of the system.
The Noether symmetries are computed for nine distinct accelerating cosmological
scenarios that contain a homogeneous scalar field associated with different
types of potentials. We verify that all the scalar field potentials, presented
here, admit the trivial first integral namely energy conservation, as they
should. We also find that the exponential potential inspired from scalar field
cosmology, as well as some types of hyperbolic potentials, include extra
Noether symmetries. This feature suggests that these potentials should be
preferred along the hierarchy of scalar field potentials. Finally, using the
latter potentials, in the framework of either quintessence or phantom scalar
field cosmologies that contain also a non-relativistic matter(dark matter)
component, we find that the main cosmological functions, such as the scale
factor of the universe, the scalar field, the Hubble expansion rate and the
metric of the FRLW space-time, are computed analytically. Interestingly, under
specific circumstances the predictions of the exponential and hyperbolic scalar
field models are equivalent to those of the $\Lambda$CDM model, as far as the
global dynamics and the evolution of the scalar field are concerned. The
present analysis suggests that our technique appears to be very competitive to
other independent tests used to probe the functional form of a given potential
and thus the associated nature of dark energy.
