  We propose to use dynamical symmetries of the field equations, in order to
classify the dark energy models in the context of scalar field (quintessence or
phantom) FLRW cosmologies. Practically, symmetries provide a useful
mathematical tool in physical problems since they can be used to simplify a
given system of differential equations as well as to determine the
integrability of the physical system. The requirement that the field equations
admit dynamical symmetries results in two potentials one of which is the well
known Unified Dark Matter (UDM) potential and another new potential. For each
hyperbolic potential we obtain the corresponding analytic solution of the field
equations. The proposed analysis suggests that the requirement of the contact
symmetry 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. Finally, in order to test the viability of the above scalar
field models we perform a joint likelihood analysis using some of the latest
cosmological data.
