  We construct a mean-field theory for itinerant ferromagnetism coexisting with
a non-unitary superconducting state, where only the majority-spin band is
gapped and contains line nodes, while the minority-spin band is gapless at the
Fermi level. Our study is motivated by recent experimental results indicating
that this may be the physical situation realized in the heavy-fermion compound
UGe$_2$. We investigate the stability of the mean-field solution of the
magnetic and superconducting order parameters. Also, we provide theoretical
predictions for experimentally measurable properties of such a non-unitary
superconductor: the specific heat capacity, the Knight shift, and the tunneling
conductance spectra. Our study should be useful for direct comparison with
experimental results and also for further predictions of the physics that may
be expected in ferromagnetic superconductors.
