Suppose a Lie group $G$ acts smoothly and properly on a manifold $X$. Does there always exist a codimension-$1$, $G$-invariant submanifold $Y$ of $X$?
Here are some thoughts on special cases when this could work.
If the action is also free, then this should be possible, provided dim $X$ is strictly greater than dim $G$. For suppose dim $M = n$ and dim $G = k<n$. Then since then the orbit space $M/G$ is a manifold of dimension $n-k$, one can take a neighbourhood $U$ of a point $x\in X/G$ that is diffeomorphic to $\mathbb{R}^{n-k}$. Take a subset $V\subseteq U$ that is diffeomorphic to $\mathbb{R}^{n-k-1}$. Then the saturation $G\cdot V$ is a codimension-$1$ submanifold of $X$.
If $G$ has finitely many connected components, it has a a maximal compact subgroup $K$. One knows in this case by a theorem of Abels that there exists a submanifold $Y\subseteq X$ such that $X$ is a trivial fibre bundle over $G/K$ with fibre $Y$, which then has codimension dim $G/K$. This makes me think that maybe it's possible to make a judicious choice of some submanifold $Y_0\subseteq X$ of codimension dim $G/K - 1$ so that the saturation $G\cdot Y_0$ has codimension $1$. But I'm not quite sure how to do this.
Any suggestions would be appreciated!