For me, they are the authentic "symmetries". They are the symmetry of a distribution corresponding to the rank 1 distribution associated with the ODE in the jet bundle.
Suppose an $n$th-order ODE
$$ u_m=\phi(x,u,\ldots,u_{m-1}) $$Generalized symmetries can be defined as vector fields
$$ Y=\xi(x, u^{(m-1)}) \partial x+\eta (x, u^{(m-1)}) \partial u + \sum_{i=1}^{m-1} \eta^{i}(x, u^{(m-1)}) \partial u_i $$with
$$ \eta^{i}(x, u^{(m-1)})=D_x(\eta^{i-1})-D_x (\xi) \cdot u_i. $$and $D_x$ the total derivative operator, and such that when prolonged one more step $Y^{(m)}$
$$ Y^{(m)}(u_m-\phi)=0 $$In @Stephani page 111 it is shown that they correspond to symmetry of a distribution of $\mathcal{S}(\{A\})$, with $A$ the associated vector field to the ODE.
A particular case are the Lie point symmetry. A more general type are nonlocal symmetrys.
In @lychagin2007contact appears the approach of generating functions.
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Author of the notes: Antonio J. Pan-Collantes
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