Symmetry group of a system of DE.

@olver86 page 93.

Let $\mathcal{S}$ be a system of DEs. A symmetry group of $\mathcal{S}$ is a local group of transformations $G$ acting on an open subset of the space of independent and dependent variables (not on the whole jet bundle associated to the trivial bundle of the variables space) of the system with the property that if $u=f(x)$ is a solution of $\mathcal{S}$ and whenever $g\cdot f$ is defined for $g\in G$ then $u=g\cdot f(x)$ is also a solution.

This group action can be prolonged to an action on the desired jet bundle of the trivial bundle made with the variables. The subvariety $\mathcal{S}$ must be invariant by the prolonged action. This is a necessary condition, but not sufficient in order to be a symmetry group for $\mathcal{S}$, since it may not transform solutions into solutions.

On the other hand, this group has infinitesimal generators (see Lie algebra) that constitute the symmetry algebra of the system. And the infinitesimal generators of the prolonged group action can be obtained from the former by means of a prolongation formula for vector fields. From this infinitesimal point of view, the condition "transform solutions into solutions" above corresponds, in essence, to this prolongation formula.
And the invariance condition is translated as: if $\mathrm{\Delta }$ is the function such that $\mathcal{S}=\{\mathrm{\Delta }=0\}$ and $X^{(n)}$ is the prolongation of (the generator of) the symmetry then

for every $p$ such that $\mathrm{\Delta }(p)=0.$

If equation (1) is satisfied for every $p$, the symmetry is called a strong symmetry.

Known facts

[Olver 1986] exercise 2.27

1. A second-order ODE admits a symmetry group of dimension at most 8. If it admits an 8-dimensional symmetry group, then it can be transformed to the ODE $u_2=0$.
2. For $n\ge 3$ a $n$th-order ordinary differential equation has at most an $(n+4)$-parameter symmetry group.
3. For system of ODEs of second order it is known the maximal dimension, but not for higher order systems.