Following @bryant2013exterior page 43, it is an exterior differential system $\mathcal{E}$ algebraic-differentially generated by 1-forms $\{\alpha_1,\ldots,\alpha_r\}$. It can be identified with the finitely generated $\mathcal{C}^{\infty}(M)$-submodule $\mathcal{P}=\mathcal{S}(\{\alpha_1,\ldots,\alpha_r\})$ of $\Omega^1(M)$ (according to @conmorando page 4 or @PaolaPfaffian), since we can recover $\mathcal{E}$ by remixing $\alpha_i$ with $\wedge$ and $d$.
The 2-forms
$$ d \alpha_i \text{ mod }(\alpha_1,\ldots,\alpha_r) $$are important in order to understand the system.
The Pfaffian system is called completely integrable when it is also algebraically generated (not only algebraic-differentially generated) by the 1-forms $\{\alpha_1,\ldots,\alpha_r\}$. In this case we have
$$ d \alpha_i=0 \text{ mod }(\alpha_1,\ldots,\alpha_r) $$From the "point of view of Paola", if we call Pfaffian system to the submodule $\mathcal{P}$, it will be completely integrable if the ideal algebraic-differentially generated by $\mathcal{P}$ (which is $\mathcal{E}$) is equal to the ideal algebraically generated by $\mathcal{P}$. Or, in the notation here, $\mathcal{P}_{alg}=\mathcal{P}_{diff}$.
Another way of saying that it is completely integrable is requiring that the derived system
$$ \mathcal{P}':=\{\beta \in \mathcal{P}:d\beta=0 \text{ mod } \mathcal{P}\} $$satisfies $\mathcal{P}'=\mathcal{P}$.
At the beginning I thought (because of the approach of Warner_1983) that a Pfaffian system was a finitely generated $\mathcal{C}^{\infty}(M)$-submodule $\mathcal{P}$ of $\Omega^1(M)$:
$$ \mathcal{P}=\{\theta^1,\ldots,\theta^s\} $$with $\theta^{\alpha}$ independent. And I thought that it was called completely integrable when the ideal $\mathcal{I}(\mathcal{P})$ of $\Omega^*(M)$ is a differential ideal. At the end, a completely integrable Pfaffian system is the same from both points of view.
They have integral manifolds.
If the Pfaffian system is generated by only one 1-form $\omega$, it will be called a Pfaffian equation.
A dual characterization of $\mathcal{P}$ is the subbundle $\mathcal{D}$ of $TM$ consisting of tangent vectors which are annihilated by $\theta^{\alpha}\in \mathcal{P}$, that is,
$$ \mathcal{D}_{x}=\left\{V \in T_{x} M:\left\langle\theta^{\alpha}, V\right\rangle=0, \quad \alpha=1, \ldots, s\right\} $$This is a distribution (see dual description of the distribution).
Consider a Pfaffian system $\mathcal{E}=\{\theta^{\alpha}\}_{diff}$ with independence condition $\Omega=\omega^1 \wedge \ldots\wedge \omega^n$. We define $J:=\{\theta^{\alpha},\omega^i\}$. The Pfaffian system will be called linear if
$$ d\theta^{\alpha}\equiv 0 \text{ mod } J $$and is usually denoted by $(I,J)$ where $I$ is the subbundle of $T^*M$ associated to $\mathcal{I}$.
The integral elements are determined by linear equations (I should work an example, see example 3 in @landsberg1997exterior.
________________________________________
________________________________________
________________________________________
Author of the notes: Antonio J. Pan-Collantes
INDEX: