Exterior differential system EDS

Definition and notation

Definition (See Bryant_2002 page xii or @bryant2013exterior page 16). An exterior differential system (EDS) is a pair $(M,\mathcal{E})$ consisting of a smooth manifold $M$ and a homogeneous differentially closed ideal $\mathcal{E}$ of the graded algebra ${\mathrm{\Omega }}^{\ast }(M)$ of differential forms on $M$.
Sometimes, an independence condition is required: a differential $n$-form $\mathrm{\Omega }$ is singled out from ${\mathrm{\Omega }}^{\ast }(M)$, in addition to $\mathcal{E}$. It is denoted $(\mathcal{E},\mathrm{\Omega })$.$◼$

Almost always (see @bryant2013exterior page 13) $\mathcal{E}$ will be generated as an algebraic and differential closure of a finite collection of differential forms (not necessarily of the same degree which, on the other hand, could be 0) ${\alpha }_A$, $1\le A\le N$. This EDSs are called of finite type.

Notation: given a set $S\subset {\mathrm{\Omega }}^{\ast }(M)$, $S_{alg}$ will denote the algebraic ideal generated by the elements of $S$. On the other hand, $S_{diff}$ will denote the ideal generated algebraic-differentially, that is, the differential closure of $S_{alg}$. Then, if $S=\{{\omega }_1,\dots ,{\omega }_k\}$

When the ideal $\mathcal{E}$ is algebraic-differentially generated by a finite set of 1-forms, we say this submodule is a Pfaffian system.

Integral manifolds

Definition. An integral manifold of an EDS $\mathcal{E}$ is a submanifold given by an immersion

such that ${\iota }^{\ast }(\phi )=0$ for all $\phi \in \mathcal{E}$.
If the EDS had an independence condition $\mathrm{\Omega }$, then ${\iota }^{\ast }(\mathrm{\Omega })$ is required to be nonvanishing.

On the other hand, an integral element of a EDS with independence condition $(\mathcal{E},\mathrm{\Omega })$ at a point $x\in M$ is $E\in G(n,T_{x}M)$ such that $\alpha {|}_E=0$ for every $\alpha \in \mathcal{E}$ and $\mathrm{\Omega }{|}_E\ne 0$. They are candidates to be tangent spaces to integral manifolds.
$◼$

An important notion is that of Cauchy characteristic vector fields.
It is also importan the notion of symmetry of an EDS and symmetry of a Pfaffian system.

Given an immersed submanifold we can define, locally, the pullback or restriction of the EDS. To define it in a global sense we need the immersion to be a closed map. See this MSE question.