The authors consider a family of perturbed Hamiltonian systems with three degrees of freedom that are in 1:1:1 resonance. The specific Hamiltonian that they actually consider has the form \[ H_\varepsilon(p,q)= H_0(p,q)+\varepsilon H_1(q)+ \varepsilon^2H_2(q), \] where \(q= (x,y,z)\), \(p=(X,Y,Z)\) and \(H_0\) is the Hamiltonian of the isotropic harmonic oscillator: \[ H_0(p,q)= {1\over 2}(x^2+ X^2)+{1\over 2}(y^2+ Y^2)+ {1\over 2}(z^2+ Z^2). \] \(H_1\) and \(H_2\) are chosen to be polynomials (cubic and quartic, respectively) that are axially symmetric with respect to the \(z\)-axis. They have the form \[ \begin{aligned} H_1(x,y,z) &= a_1z(x^2+y^2)+ a_2z^3,\\ H_2(x,y,z) &= b_1(x^2+ y^2)^2+ b_2z^2(x^2+y^2)+ b_3z^4.\end{aligned} \] The authors proceed as follows. They note that the original Hamiltonian system can be reduced using one exact symmetry (the axial one) and one approximate one (the oscillator symmetry). The oscillator symmetry can be extended up to a certain order with respect to a small parameter, and this – after truncation – allows a second reduction. The original system can also be reduced using both symmetries.
The paper proceeds by carrying out each of the three reductions. The authors analyze the three reduced systems in their corresponding reduced spaces, and then reconstruct the flow corresponding to the original Hamiltonian in all three situations. Relative equilibrium are found. The authors also include a study of stability and all the relevant parametric bifurcations.