  In this note we prove that an analytic symplectic action of a semisimple Lie
algebra can be locally linearized in Darboux coordinates. This result yields
simultaneous analytic linearization for Hamiltonian vector fields in a
neighbourhood of a common zero. We also provide an example of smooth
non-linearizable Hamiltonian action with semisimple linear part. The smooth
analogue only holds if the semisimple Lie algebra is of compact type. An
analytic equivariant b-Darboux theorem for b-Poisson manifolds and an analytic
equivariant Weinstein splitting theorem for general Poisson manifolds are also
obtained in the Poisson setting.
