  The constrained Hartree-Fock-Bogoliubov approximation, based on the recent
parametrization D1M$^{*}$ of the Gogny energy density functional, is used to
describe fission in 435 superheavy nuclei. The Gogny-D1M$^{*}$ parametrization
is benchmarked against available experimental data on inner and second barrier
heights, excitation energies of the fission isomers and half-lives in a
selected set of
  Pu, Cm, Cf, Fm, No, Rf, Sg, Hs and Fl nuclei. Results are also compared with
those obtained with the Gogny-D1M energy density functional. A detailed study
of the minimal energy fission paths is carried out for isotopic chains with
atomic numbers 100 $\le$ Z $\le$ 126 including very neutron-rich sectors up to
around 4 MeV from the two-neutron driplines. Single-particle energies, ground
state deformations, pairing correlations, two-nucleon separation energies and
barrier heights are also discussed. In addition to fission paths, the
constrained Hartree-Fock-Bogoliubov framework provides collective masses and
zero-point quantum rotational and vibrational energies. Those quantities are
building blocks within the Wentzel-Kramer-Brillouin formalism employed to
evaluate the systematic of the spontaneous fission half-lives t$_\mathrm{SF}$.
The competition between spontaneous fission and $\alpha$-decay is studied,
through the computation of the $\alpha$-decay half-lives t$_\mathrm{\alpha}$
using a parametrization of the Viola-Seaborg formula. From the comparison with
the available experimental data and the results obtained with other theoretical
approaches, it is concluded that D1M$^{*}$ represents a reasonable starting
point to describe fission in heavy and superheavy nuclei.
