  The Letter analyzes stability of spin precession currents in superfluid
$^3$He-B when the precession angle is very close to 104$^\circ$. In this limit
a spin-precession vortex has a very large core, and a barrier blocking motion
of these large-core vortices across the current streamlines (phase slip)
disappears at precession-phase gradients much smaller than critical gradients
estimated from the Landau criterion. Nevertheless, spin precession currents
remain stable up to the Landau critical gradients, since in this case there is
a barrier, which blocks the phase slip at very early stage of vortex-core
nucleation. The Letter also predicts a second-order phase transition between a
parity-symmetric and parity-nonsymmetric spin-precession vortex cores at the
precession angle 126.5$^\circ$.
