*Remark:* In this question I am first and foremost interested in a local problem and local solutions therefore I assume all functions are defined on open sets of real coordinate spaces and I will not bother with explictly considering domains. To simplify notation some maps will actually be partial maps defined on an open subset. Summation convention on repeated indices is understood. All functions are taken to be $C^\infty$.

**Background:**
A Pfaffian system is a set $\theta^\alpha=\theta^\alpha_i dx^i$ of pointwise linearly independent $1$-forms ($i=1,...,m$, $\alpha=1,...,n$, $n\le m$, i.e. we have $n$ such $1$-forms in $m$ variables).

A Pfaffian equation $$ \theta^\alpha\approx 0 $$ is a partial differential equation for a submanifold $\phi:\mathbb R^{m-n}\rightarrow\mathbb R^{m}$ which is a solution iff $$ 0=\phi^\ast\theta^\alpha=\theta^\alpha_i(\phi(u))\frac{\partial\phi^i}{\partial u^a}du^a. $$

If $\Lambda^\alpha_{\ \beta}$ is an invertible matrix whose elements are functions of the $x^i$, the system $$ \bar\theta^\alpha =\Lambda^\alpha_{\ \beta}\theta^\beta$$ is equivalent to the system $\theta^\beta$.

Per the standard terminology, the Pfaffian system is *integrable* if for each $x_0$ there is a solution (integral submanifold) whose image contains the point $x_0$. The Pfaffian system is *completely integrable* if there is an equivalent system whose generators are exact, i.e. there exists an invertible matrix $\Lambda^\alpha_{\ \beta}$ such that $$ \Lambda^\alpha_{\ \beta}\theta^\beta=dF^\alpha. $$The system is *closed* if there are $1$-forms $\xi^\alpha_\beta$ such that $$ d\theta^\alpha=\xi^\alpha_\beta\wedge\theta^\beta, $$ or equivalently if $$ d\theta^\alpha\wedge\theta^1\wedge...\wedge\theta^n=0. $$

The **Frobenius integrability theorem** essentially states that a Pfaffian system being closed is a sufficient condition for complete integrability. (The other direction of implications *completely integrable* $\Rightarrow$ *integrable* $\Rightarrow$ *closed* is easily seen.)

Assuming now $m$ "base" variables $x^i$ and $n$ "fibre" variables $y^\alpha$ and $n+m$ "total variables, a differential equation $$ \frac{\partial\phi^\alpha}{\partial x^i}(x)+\Pi^\alpha_i(x,\phi(x))=0 $$ which is often called a *total differential equation* has its existence conditions determined by the Frobenius theorem. The Frobenius condition in this case is $$ R^\alpha_{ij}=\frac{\partial\Pi^\alpha_j}{\partial x^i}-\frac{\partial\Pi^\alpha_i}{\partial x^j}+\frac{\partial\Pi^\alpha_i}{\partial y^\beta}\Pi^\beta_j-\frac{\partial\Pi^\alpha_j}{\partial y^\beta}\Pi^\beta_i=0. $$ If this equation is satisfies then for any initial condition $(x_0,y_0)$ the total differential equation has a (essentially unique) solution with $\phi^\alpha(x_0)=y_0^\alpha$. It also follows (modulo some analytical muckery regarding the smooth dependence on initial conditions) that then there exists a function $$ \Phi^\alpha(x,y_0) $$ of $m+n$ variables such that for fixed $y_0$ the function (as a function of the remaining varibles) is the unique solution corresponding to the initial values $(x_0,y_0)$, that is the general solution of the total differential equation is parametrized by $n$ constants/parameters.

**Questions:**

I am essentially looking for a *modern* (more explanation later on what I mean modern) treatment of Pfaffian systems which also take into account what happens when the integrability conditions are not satisfied identically.

For the case of a Pfaffian system $\theta^\alpha\approx 0$ and from a geometric point of view I expect that integral submanifolds of maximal ($m-n$) dimension do not exist, but lower dimensional integral manifolds might.

For total differential equations of the form $\phi^\alpha_{,i}(x)+\Pi^\alpha_{i}(x,\phi(x))=0$ I expect that a general point $(x_0,y_0)\in \mathbb R^{m+n}$ has no solution passing through it. However solutions that satisfy the algebraic constraint $$ R^\alpha_{ij}(x,\phi(x))=0 $$can still exist. I am interested in eg. how can we describe the "general solution" of the differential equation. Maybe instead of $n$ parameters, the general solution depends on $< n$ constants?

I give some further context here. I know that this problem is quite classical and its solution was even known in the 19th century. But for some reason I have extreme difficulties in finding a good and concise reference that treats this and contains the proofs.

I have searched in differential geometry books, but those usually only contain a geometric discussion of Frobenius' theorem but do not consider solutions of PDEs when the integrability conditions are not satisfied.

I am not particularly familiar with the PDE literature but looked in a few books, which only contained treatments of Laplacian/Poissonian/wave equations, boundary conditions, etc. not Pfaffians. As far as I understand, Pfaffians belong more to differential geometry than PDE theory especially that the structure of Pfaffians can be understood via ODEs.

I have found extensive discussions in *very old* books like those of Schouten/Eisenhart/Forsyth, however I have mainly three issues with these books,

- They use very archaic terminology and notations that makes reading them extremely difficult.
- It seems to me that pre-Bourbaki mathematicians had a very different idea what constitutes a proof than the post-Bourbaki mathematicians.
- They frequently assume analyticity and have often seen power series proofs in these books. I don't like that. As far as I am aware exterior differential systems more complicated than Pfaffians can only be treated in the analytic category. But the Frobenius theorem only requires $C^\infty$ (in fact $C^{\text{sufficiently high}}$) functions and thus the integrability of Pfaffian systems is treatable in the $C^\infty$ category.
*This is something I absolutely insist on, I have no interest in any treatment that uses analyticity*.

Sooooo, I'd really like to get my hands on a reasonably modern and rigorous reference on integrability conditions of Pfaffian PDEs which consider the case when the integrability conditions are not satisfied and only use $C^\infty$ arguments at most.

I also note that I am not necessarily interested in constructive solution techniques that can aid with explicitly finding the solutions/integral manifolds. I am only interested in "theoretical" characterization of integral submanifolds and/or solutions of the Pfaffian system.