Lifting smooth curves over invariants for representations of compact Lie   groups, III
  Any sufficiently often differentiable curve in the orbit space $V/G$ of a real finite-dimensional orthogonal representation $G \to O(V)$ of a finite group $G$ admits a differentiable lift into the representation space $V$ with locally bounded derivative. As a consequence any sufficiently often differentiable curve in the orbit space $V/G$ can be lifted twice differentiably. These results can be generalized to arbitrary polar representations. Finite reflection groups and finite rotation groups in dimensions two and three are discussed in detail. 
Andreas Kriegl, Mark Losik, Peter W. Michor, Armin Rainer