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We consider a scalar field which is generallly non-minimally coupled togravity and has a characteristic cubic Galilean-like term in the kinetic partof the action, in presence of a generic self-interaction as a candidate DarkEnergy model. The system is dynamically analyzed and novel fixed points withperturbative stability are demonstrated. Evolution of the system is numericallystudied near a novel fixed point which owes its existance to the Galileoncharacter of the model. It turns out that demanding the stability of this novelfixed points puts strong restriction on the allowed non-minimal coupling andthe choice of the self-interaction. The evolutions of the system is charted outon a $r-s$ diagram. The evolution of the equation of state parameter is studiedwhich shows that our model predicts accelerated universe throughout and thephantom limit is only approached closely but never crossed. Our result thusextends the findings of of \cite{Cubic_Galileon_NMC} for more general NMC thanlinear and quadratic couplings.

Non-minimally coupled quintessence DE model with a cubic galileon term --A Dynamical System Analysis