  Associated to analytic Hamiltonian vector fields in $\mathbb{C}^4$ having an
equilibrium point satisfying a non semisimple $1:-1$ resonance, we construct
two universal constants that are invariant with respect to local analytic
symplectic changes of coordinates. These invariants vanish when the Hamiltonian
is integrable. We also prove that one of these invariants does not vanish on an
open and dense set.
