  The $\eta$-pairing is a type of Cooper pairing state in which the phase of
the superconducting order parameter is aligned in a staggered manner, in
contrast to the usual BCS superconductors with a spatially uniform phase. In
this study, we search for a characteristic $\eta$-pairing state in a triangular
lattice where a simple staggered alignment of the phase is not possible. As an
example, we consider the attractive Hubbard model on both the square and
triangular lattices under strong external Zeeman field. Using the mean-field
approximation, we have identified several $\eta$-pairing states. Additionally,
we have examined the electromagnetic stability of the pairing state by
calculating the Meissner kernel. Odd-frequency pairing plays a crucial role in
achieving diamagnetic response if the electrons experience a staggered
superconducting phase during the propagation of current.
