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We know:

  1. A high order differential equation can be expressed as an ODE system.
  2. Knowledge of a symmetry allow one to reduce the order of a differential equation.

So if we do $n$-order ODE $\stackrel{2}{\longrightarrow}$ $(n-1)$-order ODE$\stackrel{1}{\longrightarrow}$ we end up with an ODE system with $n-1$ equations. This suggests we can use symmetries to reduce the number of equations in some ODE systems. This may sound trivial since we know we can use symmetries to solve ODEs but I'm looking for references, examples, etc... where one uses symmetries in a system of ODEs to obtain a smaller system (not necessarily solve the system).

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Of course, knowing a symmetry makes it possible to reduce an ODE system to a system with less components. You may wish to look into, say, Chapter 2 of the book Applications of Lie groups to differential equations by Peter Olver, for details.

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