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 $\Omega^*(M)$ of differential forms on $M$.
Sometimes, an independence condition is required: a differential $n$-form $\Omega$ is singled out from $\Omega^*(M)$, in addition to $\mathcal{E}$. It is denoted $(\mathcal{E},\Omega)$.$\blacksquare$
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) $\alpha_A$, $1\leq A\leq N$. This EDSs are called of finite type.
Notation: given a set $S\subset \Omega^*(M)$, $S_{alg}$ 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 $S=\{\omega_1,\ldots,\omega_k\}$
$$ S_{diff}=\{\omega_1,\ldots,\omega_k,d\omega_1,\ldots,d\omega_k\}_{alg} $$When the ideal $\mathcal{E}$ is algebraic-differentially generated by a finite set of 1-forms, we say this submodule is a Pfaffian system.
Definition. An integral manifold of an EDS $\mathcal{E}$ is a submanifold immersion
$$ \iota: N\to M $$such that $\iota^*(\varphi)= 0$ for all $\varphi \in \mathcal{E}$.
If the EDS had an independence condition, it is required that $\iota^*(\Omega)$ is nonvanishing.
On the other hand, an integral element of a EDS with independence condition $(\mathcal{E},\Omega)$ 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 $\Omega|_E\neq 0$. They are candidates to be tangent spaces to integral manifolds.
$\blacksquare$
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.
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Author of the notes: Antonio J. Pan-Collantes
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