Let us consider a system i) that is Hamiltonian, and ii) where we can apply the Oseledets theorem.
The presence of a symmetry ensures the presence of a vanishing (zero) Lyapunov exponent.
To be more precise: one of the Lyapunov exponents is always 0, trivially because it is associated to the time translation. There is a second vanishing Lyapunov exponent, but we get rid of it by focusing on the surfaces with fixed energy. Then, what we observe is that any additional symmetry increases the multiplicity of the exponent 0. If there is one symmetry (besides the time translation symmetry) the multiplicity of the Lyapunov exponent 0 is at least 2.
I think that this fact is trivial: please let me know if I am wrong. Btw, if you have a reference about this, it is welcome.
My question is: is the inverse also true? The inverse is:
If the multiplicity of the exponent 0 is $n>1$, then there are $n-1$ symmetries (besides the time translation symmetry).
Is this sentence true? Do you have a proof, or a counter-example?