  The main result of this article is to construct infinite families of
non-equivalent equivariant real forms of linear C*-actions on affine
four-space. We consider the real form of C* whose fixed point is a circle. In
[F-MJ] one example of a non-linearizable circle action was constructed. Here,
this result is generalized by developing a new approach which allows us to
compare different real forms. The constructions of these forms are based on the
structure of equivariant O2(C)-vector bundles.
