Generalized symmetries

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} = ϕ (x, u, \dots, u_{m - 1})

Generalized symmetries can be defined as vector fields

Y = ξ (x, u^{(m - 1)}) \partial x + η (x, u^{(m - 1)}) \partial u + \sum_{i = 1}^{m - 1} η^{i} (x, u^{(m - 1)}) \partial u_{i}

with

η^{i} (x, u^{(m - 1)}) = D_{x} (η^{i - 1}) - D_{x} (ξ) \cdot u_{i} .

and $D_{x}$ the total derivative operator, and such that when prolonged one more step $Y^{(m)}$

Y^{(m)} (u_{m} - ϕ) = 0

In @Stephani page 111 it is shown that they correspond to symmetry of a distribution of $S ({A})$ , 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.