Suppose I have a smooth distribution on an open subset $M \subset R^n$; i.e. $k$ smooth functions $X^1(\cdot),\ldots,X^k(\cdot)$ on $M$ such that, at each $p \in M$, $X^i(p) \in R^n, 1 \leq i \leq k$. Assume further that, at each $p$, $(X^1(p),\ldots,X^k(p))$ span a $k$-dimensional subspace of $R^n$. Finally, let this collection of vector fields be in involution, so that the Frobenius Theorem applies and we obtain a foliation $F$ of $M$ by integral submanifolds of dimension $k$.
My question is: Under these conditions, is each leaf $L \in F$ automatically an embedded, proper submanifold? By proper, I mean that $L$ is the image of a map $f$ from some domain in Euclidean space to $M$ such that $f^{-1}(K)$ is compact for every compact $K \subset L$.
I have tried to research this question but come up with nothing (probably because I don't know where to look). If the answer to this question is no, would things change if $k = n-1$ (i.e. if the integral submanifolds were hypersurfaces) and it were true that, for each $p \in M$, the normal to the plane spanned by $X^1(p),\ldots,X^k(p)$ was positive (meaning that each component was non-negative with at least one positive)?
Any information would be appreciated.
Edited to change immersed to embedded and to define proper in terms of compact subsets of the leaf $L$ (which, if $L$ is embedded, I guess would be intersections of compact subsets of $M$ with $L$).