A complete classification of linear foliations of $\mathbb{R}^n \setminus \{0\}$

Question

A linear $1$-form on $\mathbb{R}^n$ is a $1$-form $\alpha=\sum_i P_i(x_1,x_2,\ldots,x_n)dx_i$ such that each $P_i$ is in the linear form $P_i=\sum_j a_{ij}x_j$. A linear foliation of $\mathbb{R}^n \setminus \{0\}$ is a foliation tangent to the kernel of a linear $1$-form $\alpha$ whose corresponding matrix $(\alpha_{ij})$ is a non singular matrix.

As I learned from this answer, there are linear $1$-forms which are Frobenius integrable but are not closed $1$-form.

For $n>2$, is there a complete classification and dynamical description of all linear foliations? In particular, is there a linear foliation of $\mathbb{R}^n \setminus \{0\}$ which has a (compact) leaf with non trivial holonomy?

I think that this situation can not occurs when the corresponding $1$-form $\alpha= \sum_i \sum_j(a_{ij}x_j)dx_i $ is a closed $1$-form.(Equivalently the matrix $(a_{ij})$ is a symmetric matrix)

Answer 1

I don't know offhand the answer of your first question, but I can answer the particular situation you describe afterwards : the holonomy is always trivial.

First, notice that a compact leaf $L$ is everywhere transverse to the radial vector field $R$ (if a ray $x\mathbb R_{>0}$ were tangent to $L$ then it would be included in $L$, breaking compactness). This implies that $L$ meets any such ray exactly once : it is diffeomorphic to a $(n-1)$-sphere.

Now, every rescaling $\lambda L$ for $\lambda\in \mathbb R^\times$ is also a compact leaf, thus $\alpha$ foliates the space with smooth spheres. The foliation is therefore smoothly equivalent to the boring trivial fibration $\mathbb R_{>0}\times\mathbb S^{n-1}$.