  We investigate vortex solutions in the Ginzburg-Landau theory for neutron
$^3P_2$ superfluids relevant for neutron star cores in which neutron pairs
possess the total angular momentum $J=2$ with spin-triplet and $P$ wave, in the
presence of the magnetic field parallel to the angular momentum of vortices.
The ground state is known to be in the uniaxial nematic (UN) phase in the
absence of magnetic field, while it is in the $D_2$ ($D_4$) biaxial nematic
(BN) phase in the presence of the magnetic field below (above) the critical
value. We find that a singly quantized vortex always splits into two
half-quantized non-Abelian vortices connected by soliton(s) as a vortex
molecule with any strength of the magnetic field. In the UN phase, two
half-quantized vortices with ferromagnetic cores are connected by a linear
soliton with the $D_4$ BN order. In the $D_2$ ($D_4$) BN phase, two
half-quantized vortices with cyclic cores are connected by three linear
solitons with the $D_4$ ($D_2$) BN order. The energy of the vortex molecule
monotonically increases and the distance between the two half-quantized
vortices decreases with the magnetic field increases, except for a
discontinuously increasing jump of the distance at the critical magnetic field.
We also construct an isolated half-quantized non-Abelian vortex in the $D_4$ BN
phase.
