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)?
> 
> 
> 



what are the killing vector fields on a triaxial ellipsoid?
This follows from a rigidity theorem for convex surfaces. According to the Wikipedia page on [Cauchy rigidity](http://en.wikipedia.org/wiki/Cauchy%2527s_theorem_%2528geometry%2529), this was proved by Cohn-Vossen for smooth convex surfaces in $R^3$. So two convex surfaces which are intrinsically isometric are related by an ambient isometry of Euclidean space. So if you have a convex body in Euclidean space which has a killing field, then there would be a 1-parameter family of intrinsic isometries, and therefore a 1-parameter family of rotations. But this is clearly impossible for an ellipsoid with three different axes.


