  The present work deals with a quintom model of dark energy in the framework
of a spatially flat isotropic and homogeneous
Friedmann-Lemaitre-Robertson-Walker (FLRW) universe. At first, Lie point
symmetry is imposed to the system and the unknown coupled potential of the
model is determined. Then Noether symmetry, which is also a point like symmetry
of the Lagrangian, is imposed on the physical system and the potential takes a
general form. It is shown that the Lie algebra of Noether symmetry is a
sub-algebra of the corresponding Lie algebra of the Lie symmetry. Finally, a
point transformation in the three dimensional augmented space is performed
suitably so that one of the variables become cyclic and as a result there is
considerable simplification to the physical system. Hence conserved quantities
(i.e, constants of motion) are expressed in a compact form and cosmological
solutions are evaluated and analyzed in the present context.

General Relativity and Quantum Cosmology