Equivariant affine line bundles and linearization

We show that every algebraic action of a linearly reductive group on affine n-space C^n which is given by Jonqui`ere automorphisms is linearizable. Similarly, every holomorphic action of a compact group K by (holomorphic) Jonquière automorphisms is linearizable. Moreover, any holomorphic action of K on C^2 by overshears is linearizable, too. These results are based on the fact that equivariant algebraic or holomorphic affine line bundles over C^n are trivial.