  We focus on the existence and persistence of families of saddle periodic
orbits in a four-dimensional Hamiltonian reversible ordinary differential
equation derived using a travelling wave ansatz from a generalised nonlinear
Schr{\"o}dinger equation (GNLSE) with quartic dispersion. In this way, we are
able to characterise different saddle periodic orbits with different signatures
that serve as organising centres of homoclinic orbits in the ODE and solitons
in the GNLSE. To achieve our objectives, we employ numerical continuation
techniques to compute these saddle periodic orbits, and study how they organise
themselves as surfaces in phase space that undergo changes as a single
parameter is varied. Notably, different surfaces of saddle periodic orbits can
interact with each other through bifurcations that can drastically change their
overall geometry or even create new surfaces of periodic orbits. Particularly
we identify three different bifurcations: symmetry-breaking, period-$k$
multiplying, and saddle-node bifurcations. Each bifurcation exhibits a
degenerate case, which subsequently gives rise to two bifurcations of the same
type that occurs at particular energy levels that vary as a parameter is
gradually increased.
  Additionally, we demonstrate how these degenerate bifurcations induce
structural changes in the periodic orbits that can support homoclinic orbits by
computing sequences of period-$k$ multiplying bifurcations.
