A generalization of Puiseux's theorem and lifting curves over invariants

Let $\rho: G \to \operatorname{GL}(V)$ be a rational representation of a reductive linear algebraic group $G$ defined over $\mathbb C$ on a finite dimensional complex vector space $V$. We show that, for any generic smooth (resp. $C^M$) curve $c : \mathbb R \to V // G$ in the categorical quotient $V // G$ (viewed as affine variety in some $\mathbb C^n$) and for any $t_0 \in \mathbb R$, there exists a positive integer $N$ such that $t \mapsto c(t_0 \pm (t-t_0)^N)$ allows a smooth (resp. $mathbb C^M$) lift to the representation space near $t_0$. ($C^M$ denotes the Denjoy--Carleman class associated with $M=(M_k)$, which is always assumed to be logarithmically convex and derivation closed). As an application we prove that any generic smooth curve in $V // G$ admits locally absolutely continuous (not better!) lifts. Assume that $G$ is finite. We characterize curves admitting differentiable lifts. We show that any germ of a $C^\infty$ curve which represents a lift of a germ of a quasianalytic $C^M$ curve in $V // G$ is actually $C^M$. There are applications to polar representations.