Existence of a certain foliation of $\mathbb R^n$ Notation:
We say two $C^1$ manifolds are $C^1$-homeomorphic if they are homeomorphic via a $C^1$ homeomorphism with $C^1$ inverse.
Question:
Let $n \geq 2$. Given a countable dense set of points $P \subset \mathbb R^n$, does there exist a $C^1$ foliation $M_{\alpha}$ of $\mathbb R^n$ by $C^1$ manifolds that are $C^1$-homeomorphic to $\mathbb R^{n-1}$ with the following property?
Suppose the foliation $M_{\alpha}$ is parametrised by $\alpha \in (0, 1)$. Then the set of all $\alpha$ such that $M_{\alpha} \cap P$ is dense in $M_{\alpha}$ is dense in $(0, 1)$.
 A: EDIT: Originally I could prove that there is such a foliation by topological manifolds:
Clearly, if $\mathbb{Q}^n$ is the set if points with all rational coordinates, you can have a foliation by parallel hyperplanes $H_\alpha$. Now, for any set $P$ there that is countable and dense there is homepmorphism $\Phi:\mathbb{R}^n\to\mathbb{R}^n$ such that $\Phi(\mathbb{Q}^n)=P$, see Claim A) on page 44 in [1]. Then $M_\alpha=\Phi(H_\alpha)$ is a foliation by topological manifolds.
However, as pointed out by Alessandro Codenotti in a comment below, Morayne proved in [2]  that there is $\Phi$ with the properties listed above that is a volume preserving analytic diffeomorphism so in fact the foliation can be done by analytic manifolds.
[1] W. Hurewicz, H. Wallman, Dimension Theory.
Princeton Mathematical Series, vol. 4. Princeton University Press, Princeton, N. J., 1941.
[2] M. Morayne,
Measure preserving analytic diffeomorphisms of countable dense sets in $C^n$ and $R^n$. Colloq. Math. 52 (1987), 93–98.
