Lifting differentiable curves from orbit spaces

Let $\rho : G \rightarrow \operatorname{O}(V)$ 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 $d = \max_i \operatorname{deg} \sigma_i$. We prove that for each $C^{d-1,1}$-curve $c$ in $\sigma(V) \subseteq \mathbb R^n$ there exits a locally Lipschitz lift over $\sigma$, i.e., a locally Lipschitz curve $\overline c$ in $V$ so that $c = \sigma \circ \overline c$, and we obtain explicit bounds for the Lipschitz constant of $\overline c$ in terms of $c$. Moreover, we show that each $C^d$-curve in $\sigma(V)$ admits a $C^1$-lift. For finite groups $G$ we deduce a multivariable version and some further results.