{\bf Abstract} The steady bifurcation flows in a spherical gap
(gap ratio $\sigma=0.18$) with rotating inner and stationary outer spheres
 are simulated numerically for $Re_{c_1}\leq Re\leq 1500$ by solving
the steady axisymmetric incompressible Navier--Stokes equations  using a finite
 difference method. The simulation shows that two stable steady  flows
with 1 or 2 vortices per hemisphere exist for $775 \leq Re \leq 1220$ while
three stable steady flows with 0, 1, or 2 vortices per hemisphere
 exist for $ 1220<Re\leq1500$.
The formation of different flows at the same Reynolds number is related
with different initial conditions which can be generated by different
accelerations of the inner sphere. Generation of the zero-- or two--vortex
flow depends mainly on the acceleration while that of the one--vortex flow also
depends on a type of perturbation which breaks the  equatorial  symmetry.
The mechanism of development of a saddle point in the meridional plane at
higher $Re$ number and its role in the formation of two--vortex flow
are analyzed.

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{\bf Keywords: numerical simulation, spherical
Taylor-Couette flow, non-unique solutions of the N-S equations,
symmetry-breaking bifurcation}