  Kane fermions are characterized by a linear Dirac cone intersecting with a
flat band, resembling a pseudo-spin-1 Dirac semimetal. Similar to relativistic
Dirac fermions, Kane fermions satisfy a linear energy-momentum relation and can
be classified as being pseudo-relativistic. Though not protected by symmetry or
by topology, Kane fermions can emerge by suitable band engineering, for
example, in mercury-telluride compounds. Here we study the Berry curvature of
Kane fermions that emerges in the presence of time-reversal symmetry breaking
weak Zeeman fields. We discuss the related anomalous transport coefficients and
discuss the anisotropy in these responses that can be probed in experiments.
