  We present an analysis of stationary solutions for two-dimensional (2D)
Bose-Einstein condensates (BECs) with the Rashba spin-orbit (SO) coupling and
Zeeman splitting. By introducing the generalized momentum operator, the linear
version of the system can be solved exactly. The solutions are semi-vortices of
the Bessel-vortex (BV) and modified Bessel-vortex (MBV) types, in the presence
of the weak and strong Zeeman splitting, respectively. The ground states (GSs)
of the full nonlinear system are constructed with the help of a specially
designed neural network (NN). The GS of the mixed-mode type appears as
cross-attraction interaction increases. The spin texture of the GS is produced
in detail. It exhibits the Neel skyrmion structure for the semi-vortex GS of
the BV type, and the respective skyrmion number is found in an analytical form.
On the other hand, GSs of the MBV and mixed-mode types do not form skyrmions.
