J Ize conjectured that for any absolutely irreducible representation of a compact Lie group G on a finite dimensional real vectorspace there exists an isotropy subgroup which has an odd dimensional fixed point space. If it were true it had immediate consequences in equivariant bifurcation. Lauterbach & Matthews showed that this is not the case. Their findings of three infinite families of finite groups were supplemented by extensive computer analysis showing a very difficult zoo of groups acting on R4. In this talk we will give a complete list of counter examples in R4. In our notation we follow the group theoretical notation by Conley & Smith.
We discuss the open question of infinite compact Lie groups as counter examples to the Ize conjecture. Finally we give an overview on the bifurcation scenario and an outlook on higher space dimensions.