  Ising spin-orbital coupling is usually easy to identify in the Ising
superconductors via an in-plane critical field enhancement, but we show that
the Ising spin-orbital coupling also manifests in the vortex physics for
perpendicular magnetic fields. By self-consistently solving the Bogoliubov-de
Gennes equations of a model Hamiltonian built on the honeycomb lattice with the
Ising spin-orbital coupling pertinent to the transition metal dichalcogenides,
we numerically investigate the local breaking of the spin and sublattice
degeneracies in the presence of a perpendicular magnetic field. It is revealed
that the ferromagnetic orders are induced inside the vortex core region by the
Ising spin-orbital coupling. The induced magnetic orders are antiphase in terms
of their opposite polarizations inside the two nearest-neighbor vortices with
one of the two polarizations coming dominantly from one sublattice sites,
implying the local breaking of the spin and sublattice degeneracies. The
finite-energy peaks of the local-density-of-states for spin-up and spin-down
in-gap states are split and shifted oppositely by the Ising spin-orbital
coupling, and the relative shifts of them on sublattices $A$ and $B$ are also
of opposite algebraic sign. The calculated results and the proposed scenario
may not only serve as experimental signatures for identifying the Ising
spin-orbital coupling in the Ising superconductors, but also be prospective in
manipulation of electron spins in motion through the orbital effect in the
superconducting vortex states.
