  This paper is devoted to studying Hamiltonian oscillators in 1:1:1:1
resonance with symmetries, which include several models of perturbed Keplerian
systems. Normal forms are computed in Poisson and symplectic formalisms, by
mean of invariants and Lie-transforms respectively. The first procedure relies
on the quadratic invariants associated to the symmetries, and is carried out
using Gr\"obner bases. In the symplectic approach, hinging on the maximally
superintegrable character of the isotropic oscillator, the normal form is
computed {\it a la} Delaunay, using a generalization of those variables for
4-DOF systems. Due to the symmetries of the system, isolated as well as circles
of stationary points and invariant tori should be expected. These solutions
manifest themselves rather differently in both approaches, due to the
constraints among the invariants versus the singularities associated to the
Delaunay chart.
  Taking the generalized van der Waals family as a benchmark, the explicit
expression of the Delaunay normalized Hamiltonian up to the second order is
presented, showing that it may be extended to higher orders in a
straightforward way. The search for the relative equilibria is used for
comparison of their main features of both treatments. The pros and cons are
given in detail for some values of the parameter and the integrals.
