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We consider a dynamical system moving in a Riemannian space and prove two theorems which relate the Lie point symmetries and the Noether symmetries of the equation of motion, with the special projective group and the homothetic group of the space respectively. These theorems are used to classify the two dimensional Newtonian dynamical systems, which admit Lie point/Noether symmetries. The results of the study i.e. expressions of forces / potentials, Lie symmetries, Noether vectors and Noether integrals are presented in the form of tables for easy reference and convenience. Two cases are considered, Hamiltonian and non-Hamiltonian systems. The results are used to determine the Lie / Noether symmetries of two different systems. The Kepler -Ermakov system, which in general is non-conservative, and the conservative system with potential similar to the Hènon Heiles potential. As an additional application, we consider the scalar field cosmologies inFRW background with no matter, and look for the scalar field potentials for which the resulting cosmological models are integrable. It is found that the only integrable scalar field cosmologies are defined by the exponential and the Unified Dark Matter potential. It is to be noted that in all aforementioned applications the Lie / Noether symmetry vectors are found by simply reading the appropriate entry in the relevant tables. 1 Of course it is possible to look for a metric for which a given set of Γ i jk are the connection coefficients, or, even avoid the metric altogether. However we shall not attempt this in the present work. For such an attempt see [7].

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.Comment: Accepted for publication in Physical Review D (13 pages

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