Lifting differentiable curves from orbit spaces

Let $\rho :G\to \mathrm{O}(V)$ be a real finite dimensional orthogonal representation of a compact Lie group, let $\sigma = (\sigma_1,\ldots,\sigma_n) : V \to \mathbb R^n$,where$\sigma_1,\ldots,\sigma_n$ form a minimal system of homogeneous generators of the $G$-invariant polynomials on $V$, and set $d = \max_i \operatorname{deg} \sigma_i$\mathrm{.}Weprovethatforeach$C^{d-1,1}$-curve$c$in$\sigma(V) \subseteq \mathbb R^n$thereexitsalocallyLipschitzliftover$\sigma$, i.e., a locally Lipschitz curve $\bar{c}$ in $V$ so that $c = \sigma \circ \overline c$,andweobtainexplicitboundsfortheLipschitzconstantof$\overline c$intermsof$c$\mathrm{.}Moreover,weshowthateach$C^d$-curvein$\sigma(V)$admitsa$C^1$-lift\mathrm{.}Forfinitegroups$G$ we deduce a multivariable version and some further results.