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.
Gutwirth,The action of an algebraic torus on the affine plane, Trans
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Criteria are given for local finiteness of solvable Lie algebras of derivations on affine varieties, with an affirmative answer for the affine plane under an additional assumption.
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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.
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Locally finite solvable Lie algebras of derivations
Criteria are given for local finiteness of solvable Lie algebras of derivations on affine varieties, with an affirmative answer for the affine plane under an additional assumption.