  In this work, we examine solutions of the system of equations obtained by
applying the Noether gauge symmetry (NGS) and its conserved quantity for the
standard general relativity (GR) and the non-minimal derivative coupling (NMDC)
cosmological model. We discover two salient features of the solutions. The
first one is $a(t)\propto t^{1/3}$ for a kinetic-dominant phase which may
emerge before inflationary period at very early time. The second one is a new
form of scalar field $\phi(t)$ govern by the exponential cosmological solution
for GR and NMDC $\phi_{\rm GR}(t)= \sqrt{c_{1}+c_{2}t+c_{3}e^{-\lambda t}} $
and $\phi_{\rm NMDC}(t)=
\sqrt{c_{1}+c_{2}e^{-\lambda_{1}t}+c_{3}e^{-\lambda_{2}t}}$, respectively.
