  It has been generally accepted that the spin-orbit coupling effect in
noncentrosymmetric materials leads to the band splitting and non-trivial spin
polarization in the momentum space. However, in some cases, zero net spin
polarization in the split bands may occurs, dubbed as the band splitting with
vanishing spin polarization (BSVSP) effect, protected by non-pseudo-polar point
group symmetry of the wave vector in the first Brillouin zone [Liu et. al.,
Nat. Commun. \textbf{10}, 5144 (2019)]. In this paper, by using
first-principles calculations, we show that the BSVSP effect emerges in
two-dimensional (2D) nonsymmorphic Ga$XY$ ($X$= Se, Te; $Y$= Cl, Br, I) family,
a new class of 2D materials having in-plane ferroelectricity. Taking the GaTeCl
monolayer as a representative example, we observe the BSVSP effect in the split
bands along the $X-M$ line located in the proximity of the conduction band
minimum. By using $\vec{k}\cdot\vec{p}$ Hamiltonian derived based on the
symmetry analysis, we clarify that such effect is originated from the
cancellation of the local spin polarization, enforced by non-pseudo-polar
$C_{2v}$ point group symmetry of the wave vector along the $X-M$ line.
Importantly, we find that the spin polarization can be effectively induced by
applying an external out-of-plane electric field, indicating that an
electrically tunable spin polarization for spintronic applications is
plausible.
