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

Nematic and time-reversal breaking superconductivities coexisting with quadrupole order in a $\Gamma_3$ system