Suppose I have a smooth (say $C^1$) codimension one foliation of $P^n$ (open subset of $R^n$ consisting of vectors with all positive components) arising from a smooth $(n-1)$-plane field satisfying the Frobenius conditions. Now take a single leaf $L$; I would like to show that $L$ is the graph of a function defined on some open subset of $P^{n-1}$. If I knew that $L$ was the level set of some function $F$ on $P^n$, say $F(x_1,\ldots,x_n) = 0, \forall x \in P^n$, and I knew that $F_n(p) > 0 \; \forall p \in L$, then I could use the "basic" implicit function theorem to conclude that there exists a function $u$ such that $F(x_1,\ldots,x_{n-1},u(x_1,\ldots,u_{n-1})) = 0$ locally, then proceed to extend it since I know $F_n(p) > 0$ everywhere on $L$.
My question is: Do I know that $L$ is the level set of some function $F$? For reasons outside the scope of this question I do know that the normal vector to the tangent plane to $L$, at every point of $L$, has $n$th component $> 0$ (I can compute this manually since I know the vector fields giving rise to the distribution), but to me that doesn't seem to make things directly amenable to an application of the implicit function theorem. I have a feeling that there is some way to look at this which makes the answer to my question very obvious (and the question itself silly), but I have tried to work this out and being very obsessive about my proofs I wanted to ask here.
Any info or advice appreciated.
Edit to add information.: I understand that $L$, being an integral manifold, is a graph, locally, say on some domain $U$. I believe my condition that the last component of the normal to the tangent plane, at every point $p \in L$, means that $U$ can be extended to a maximal domain $D$ by means of the implicit function theorem. I just am searching for the proper application of that theorem given that I don't know that $L$ can be given by a function of the form $F(x_1,\ldots,x_n) = 0$.
Edit to clarify my objective I want to show that any leaf $L$ of a particular foliation (arising from a specific $n$-plane distribution on $P^n$) is the graph of some function $u : D \rightarrow P$ where $D$ is an open subset of $P^{n-1}$. This specific distribution gives me the condition on the normals mentioned in the previous edit. I believe the way to prove that is to show that any neighborhood $U$ on which we know $L$ is a graph (there has to be one since an integral manifold is locally a graph) can always be extended to a maximal domain unless $u$ goes to infinity or zero on its boundary, and I believe the way to do that is via the implicit function theorem, utilizing the condition on the normals. I just don't know how to prove it this way.
Edit to give specifics: Let $n=3$; I have two strictly positive, $C^2$ functions on $P^3$, $m^1(x,y,z)$ and $m^2(x,y,z)$. The vector fields are $X^1 = \partial/\partial x - m^1 \partial/\partial z$ and $X^2 = \partial/\partial y - m^2 \partial/\partial z$ and I impose the following condition on the derivatives of $m^1,m^2$: $ -\partial m^2/\partial x + \partial m^1/\partial y + m^1 \partial m^2/\partial z - m^2 \partial m^1/\partial z = 0$. This should make the $2$ plane distribution generated by $X,Y$ involutive and so by Frobenious there is a smooth foliation $\mathscr{F}$ of $P^3$ by integral manifolds of the distribution. The normal vector to the plane tangent to a leaf $L$ at a point $p$ can be computed as $\pm(m^1(p),m^2(p),1)$, I take the positive normal. My understanding is that under these conditions (in particular, that the normal has strictly positive last component at every point of $P^3$) each leaf $L \in \mathscr{F}$ is the graph of some function $u : D \rightarrow P$, where $D$ is some open set of $P^2$. I am looking for details on proving (if possible) my understanding, and my idea was to use the implicit function theorem.