Rank and rigidity of locally nilpotent derivations of affine fibrations

In this exposition, we propose a notion of rank and rigidity of locally nilpotent derivations of affine fibrations. We show that the concept is analogous to the perception of rank and rigidity of locally nilpotent derivations of polynomial algebras. Our results characterize locally nilpotent derivations of ${\mathbb{A}}^3$-fibrations having slice by classifying the fixed point free locally nilpotent derivations in terms of their ranks.