Generating function

In @lychagin2007contact page 32 it is said that for an ODE

u_{m} = F (x, u, \dots, u_{m - 1})

the generalized symmetries $S$ of it, when transformed into an evolutionary vector field can be written like

S = ϕ \partial u + A (ϕ) \partial u_{1} + \dots + A^{k} (ϕ) \partial u_{m - 1}

The function $ϕ$ is called generating function. A vector field like $S$ is a generalized symmetry of an ODE with vector field $A$ if and only if Lie equation

A^{m} (ϕ) - \sum_{i = 0}^{m - 1} \frac{\partial F}{\partial u_{i}} A^{i} (ϕ) = 0

is satisfied.

The geometric interpretation of this is as follows. A solution $f (x)$ of the equation can be seen, instead of as a curve in the jet space, as a point in the vector space $C^{\infty} (M)$ . A symmetry (well, its evolutionary vector field) produces a curve that starts at $f (x)$ and every point on it is another solution. The tangent vector of that curve is $ϕ$ (see remark 2.1.1 in @lychagin2007contact).

This approach suggests to me the issue of the connected components of the solution space of an ODE.