Symmetry of an exterior differential system

Given a exterior differential system $I$ on a manifold $M$ , a vector field $X \in X (M)$ is called a symmetry of $I$ if

L_{X} ω \in I

for every $ω \in I$ .

Remarks
(see @ivey2016cartan exercises 6.1.2)

Symmetries of an EDS form a Lie algebra.
To show that $X$ is a symmetry we only need to check the condition above in a set of form which generate $I$ differentially (see formulas for Lie derivative, exterior derivatives, bracket, interior product).

A particular case of symmetries is given by the Cauchy characteristic vector fields.

Theorem (Th 2.3.3 Barco thesis). Given an ideal $I$ and suppose that the Cauchy characteristic space $A (I)$ is not the 0 modulus. If a vector field $X$ is a symmetry of $I$ then it is also a symmetry of the distribution $A (I)$ . $◼$

Related: symmetry of a Pfaffian system

Example – Integration using a single infinitesimal symmetry

Consider the second‑order PDE of F‑Gordon type

u_{x y} = \frac{\sqrt{u_{x} u_{y}}}{x + y}

The associated exterior differential system $I$ on the manifold $M$ with coordinates $(x, y, u, u_{x}, u_{y})$ is generated by the contact form $θ = d u - u_{x} d x - u_{y} d y$ and the 2‑form encoding the PDE. The system admits the translational infinitesimal symmetry $X = \partial_{u}$ (generated by the group $u \mapsto u + c$ ). Reducing by this symmetry (i.e. passing to the quotient $\bar{M} = M / R$ with $u$ eliminated) yields the reduced EDS $\bar{I}$ on $\bar{M}$ with coordinates $(x, y, v, w)$ , where $v = u_{x}$ , $w = u_{y}$ , generated by

d x \land d v + d y \land d w, (d v - \sqrt{v w} \frac{d y}{x + y}) \land d x .

The reduced system is equivalent to the first‑order equations

v_{y} = \frac{\sqrt{v w}}{x + y}, w_{x} = \frac{\sqrt{v w}}{x + y} .

These equations can be integrated directly. The original solution is then recovered by integrating the Lie‑type equation for the group parameter associated to $X$ , which in this one‑dimensional case reduces to a quadrature. Hence the original EDS is integrated by first solving the reduced system and then lifting via the infinitesimal symmetry $X$ .

Source: Fels_EDS notebook