  We study the scattering rigidity problem for standard stationary manifolds
using timelike geodesics with a fixed momentum. Taking advantage of the
symmetry of this manifolds, we use Hamiltonian reduction to show that this
problem is related to scattering rigidity for $\mathcal{MP}$-systems, a problem
studied before. This gives several new rigidity results (up to some gauge) for
this kind of Lorentzian manifolds.
