  In this article, the modified Kane-Mele Hamiltonian is derived for graphene
with wedge disclination and spin orbit couplings (intrinsic and Rashba). The
wedge disclination changes the flat lattice into the conical lattice and hence
modifies the spin orbit couplings. The Hamiltonian is exactly solved for the
intrinsic spin orbit interaction and perturbatively for the Rashba spin orbit
interaction. It is shown that there exists the Kramer's degenerate midgap
localized spin separated fluxon states around the defect. These zero energy
spin separated states occur at the external magnetic flux value
$\Phi\pm\Delta\Phi$. The external magnetic flux $\Phi$ is introduced to make
the wave-function periodic when the electron circulates around the defect. It
is found that this separation occurs due to the effect of the conical curvature
on the spin orbit coupling. Further, we find these results are robust to the
addition of the Rashba spin orbit interaction which is important for the
application to spintronics and nanoelectronics.
