Reading a paper on hamiltonian mechanics, in a section on classical examples of complete integrability, it is examined the geodesic flow of a triaxial ellipsoid.

Before separating the variables in the Hamilton-Jacobi equation, the original approach followed by Jacobi himself, it is stated that the phase space of this system has none infinitesimal symmetry which is the lift of vector field on the space of the configuration.

So this would be an example of an hamiltonian system of mechanical type with only hidden (or dynamical) symmetries.

My question is: how to prove that there are not Killing vector fields for the triaxial ellipsoid (endowed of the ambient euclidean metric)?