In @lychagin2007contact page 32 it is said that for an ODE
$$ u_m=F(x,u,\ldots,u_{m-1}) $$the generalized symmetries $S$ of it, when transformed into an evolutionary vector field can be written like
$$ S=\phi {\partial u}+A(\phi) {\partial u_{1}}+\cdots+A^{k}(\phi) {\partial u_{m-1}} $$The function $\phi$ 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
$$ A^{m}(\phi)-\sum_{i=0}^{m-1} \frac{\partial F}{\partial u_{i}} A^{i}(\phi)=0 $$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 $\mathcal{C}^{\infty}(M)$. 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 $\phi$ (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.
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Author of the notes: Antonio J. Pan-Collantes
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