  We discuss superconductivity in a model on a cubic lattice for a $\Gamma_3$
non-Kramers system. In previous studies, it is revealed that $d$-wave
superconductivity with $E_g$ symmetry occurs in a wide parameter range in a
$\Gamma_3$ system. Such anisotropic superconductivity can break the cubic
symmetry of the lattice. In a $\Gamma_3$ system, the quadrupole degrees of
freedom are active and the effect of the cubic symmetry breaking should be
important. Here, we investigate the coexisting states of the $d$-wave
superconductivity and quadrupole order by a mean-field theory. In particular,
we discuss possible competition and cooperation between the superconductivity
and quadrupole order depending on types of them. We find nematic
superconductivity breaking the cubic symmetry and coexisting with quadrupole
order. In the present model, we also find $d+id$ superconductivity, which
breaks time-reversal symmetry but retains the cubic symmetry. We also discuss
the effects of uniaxial stress on these superconducting states.
