  In cosmological framework, Noether symmetry technique has revealed a useful
tool in order to examine exact solutions. In this work, we first introduce the
Jordan-frame Lagrangian and apply the conformal transformation in order to
obtain the Lagrangian equivalent to Einstein-frame form. We then analyse the
dynamics of the field in the cosmological alpha-attractors using the Noether
sysmetry approach by focusing on the single field scenario in the
Einstein-frame form. We show that with a Noether symmetry the coresponding
dynamical system can be completely integrated and the potential exhibited by
the symmetry can be exactly obtained. With the proper choice of parameters, the
behavior of the scale factor displays an exponential (de Sitter) behavior at
the present epoch. Moreover, we discover that the Hubble parameters strongly
depends on the initial values of parameters exhibited by the Noether symmetry.
Interestingly, it can retardedly evolve and becomes a constant in the present
epoch in all cases.
