term $$f^{(2)}_{Z} = -\tilde{\kappa} h (|\Delta_{\uparrow\!\uparrow}|^2 -
|\Delta_{\downarrow\!\downarrow}|^2),$$ plays an important role for the HQV vortex lattice by introducing a slight spin-polarization. This slight spin-polarization gives rise to two phase transitions as in the case of the A$_1$/A$_2$ phase of $^3$He[@RevModPhys.47.331; @VollhardtHe3]. The inhomogeneous part of the quadratic terms $f^{(2)}_{in}$ are of the form $$K_{ij;kl} (D_i \Delta_k) (D_j \Delta_l)^* + {\rm c.c.},$$ where $D_i = \nabla_i + (2\pi i/\Phi_0)A_i$. For these terms, we require rotational invariance up to the lattice symmetry with respect to orbital degrees of freedom only, which means that we require invariance with respect to rotating $D_i$’s and the orbital component of $\Delta_i$’s. The complex structure of $f^{(2)}_{in}$ can result in the condensate wave function of a different form than that of conventional SC. $f^{(2)}_{SO}$ is the quadratic spin-orbit coupling term assuming the spin-orbit coupling to be small and is discussed in Sec. \[sec:SO\]. In the presence of spin-orbit coupling the free energy have to be invariant under the combined discrete rotation of the orbital and spin degree of freedom specific for the given lattice symmetry. For lattices with orthogonal or tetragonal symmetry, spin-orbit coupling may reduce the symmetry of the Gibbs free energy and tend to 