Let $\Upsilon: G \times M \to M$ be a smooth action of a Lie group $G$ on a manifold $M$.

Isenberg and Marsden (1982) define a slice at $m \in M$ as a submanifold $S \subseteq M$ containing $m$ such that

- $S$ is invariant under the induced action of $G_m$.
- $(\Upsilon_g S) \cap S \neq \emptyset$ implies $g \in G_m$.
- There exists a local section $\chi: G/G_m \supseteq U \to G$ defined in a neighborhood $U$ of the identity coset such that the map \begin{equation} \chi^S: U \times S \to M, \qquad ([g], s) \mapsto \chi([g]) \cdot s \end{equation} is a diffeomorphism onto a neighborhood $V \subseteq M$ of $m$.

The same definition can be found in KrieglMichor1997, with a short comment that the usual finite-dimensional definition is too rigid for infinite-dimensional applications. What are concrete examples where this weaker definition of a slice is required and the usual finite-dimensional version does not apply?