Generalized ${\mathcal{C}}^{\mathrm{\infty }}$-symmetry of ODE

These symmetries play a role analogous to cinf-symmetry of distribution but in the specific case of an ODE (which is simply a particular case of a distribution spanned by a single vector field $A$). That is, they satisfy the condition:

for some smooth functions $\lambda$ and $\mu$.

They do not correspond exactly to the ${\mathcal{C}}_k^{\mathrm{\infty }}$-symmetries of C. Muriel et al., which would be an even more particular case. These are considered a "slight generalization" of those concepts.

However, they adjust to the same prolongation formula, meaning they can often be found using similar determining equations.

These vector fields satisfy the IBD property for vectors.