Algebraic groups of automorphisms of polynomial rings

Some general results on algebraic group actions, with a focus on linearizability, are applied to constructing nonlinearazable actions of nonsolvable nonreductive groups with fixed points and to proving that any algebraic action of a connected reductive algebraic group $G$ on $n$-dimensional affine space over an algebraically closed field of characteristic zero is linearizable in either of the cases:
(1) $n=3$;
(2) $n=4$ and $G$ is not a one or two dimensional torus.