A locally nilpotent derivation D with slice s yields a G_m-action via the semisimple derivation NsD whose linearizability in the nice case holds exactly when D is conjugate to a partial derivative and the slice is affine-linear.
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On nice $\mathbb{G}_m$-actions arising from locally nilpotent derivations with slice
A locally nilpotent derivation D with slice s yields a G_m-action via the semisimple derivation NsD whose linearizability in the nice case holds exactly when D is conjugate to a partial derivative and the slice is affine-linear.