Let $f:\mathbb{R}\times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ be a smooth function and $G\subset SO(n)$ be a $1-$dimensional compact Lie Group (diffeomorphic to the circle). Moreover let $G$ act on $\mathbb{R}^{n}$ by standard left multiplication. We assume that $f$ is equivariant with respect to $G$, i.e. for all $g\in G$ and all $(t,x)\in \mathbb{R}\times \mathbb{R}^{n}$ we have $f(t,g\cdot x) = g\cdot f(t,x)$. Let now $(t_{0},x_{0}) \in \mathbb{R}\times \mathbb{R}^{n}$ such that $x_{0} \not= 0$, $f(t_{0},x_{0}) = 0$ and

$$\ker \left ( \frac{\partial f}{\partial x}(t_{0},x_{0}) \right ) = T_{x_{0}}(G\cdot x_{0}),$$ i.e. the kernel of the Jacobian of $f$ is only in the direction of the action.

QUESTION: Is there any version of the implicit function theorem in this setting? Does one need more additional conditions to be able to construct near-by solutions? If yes, which conditions are these?

Thanks in advance. Hope for answers. Greetings, Ben

Let $f:\mathbb{R}\times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ be a smooth function and $G\subset \operatorname{SO}(n)$ be a $1$-dimensional compact Lie group (diffeomorphic to the circle). Moreover let $G$ act on $\mathbb{R}^{n}$ by standard left multiplication. We assume that $f$ is equivariant with respect to $G$, i.e. for all $g\in G$ and all $(t,x)\in \mathbb{R}\times \mathbb{R}^{n}$ we have $f(t,g\cdot x) = g\cdot f(t,x)$. Let now $(t_{0},x_{0}) \in \mathbb{R}\times \mathbb{R}^{n}$ such that $x_{0} \not= 0$, $f(t_{0},x_{0}) = 0$ and

$$\ker \left ( \frac{\partial f}{\partial x}(t_{0},x_{0}) \right ) = T_{x_{0}}(G\cdot x_{0}),$$ i.e. the kernel of the Jacobian of $f$ is only in the direction of the action.

QUESTION: Is there any version of the implicit function theorem in this setting? Does one need more additional conditions to be able to construct near-by solutions? If yes, which conditions are these?

Thanks in advance.