  We get point vortices dynamics equations on a rotating sphere surface
directly from the hydrodynamic equations as representing their weak exact
solution contrary to the conventional case of the use of a kinematic
relationship between a given singular vortex field and velocity field. It is
first time that the effect of a sphere rotation on the vortices interaction is
accounted for in exact form. We show that only the stream function of a vortex
pair of antipodal vortices (APV), and only it satisfies the original
three-dimensional hydrodynamics equations on a sphere. We prove that only APV
pair with two point vortices in the diameter-conjugated points of a sphere with
equal by quantity but different sign circulations may be correctly considered
as an elementary (stationary, not self-affecting) singular point object on a
sphere. We suggest using the axis connecting the two point vortices in an APV
for describing of an axis of rotation of the global vortices introduced in
(Barrett, 1958) to reflect the observed global rotation of atmospheric masses
with the rotation axes not coinciding with the planet rotation axis and
precessing about it. Up to now, the question about interaction of global
vortices corresponding to such solutions with rotations about different axis
was not even posed. This is the first model describing interaction of the
Barrett-type global vortices corresponding to atmospheric centers of action
(ACA). The new steady-state and its stability conditions for N=2 are obtained
and used for the analysis of coupled cyclone-anticyclone ACAs over oceans in
the Northern Hemisphere. On the base of corresponding exact solutions, we show
acceptability of modelling of the stable blocks of splitting flow type when in
line with the exact accounting for the sphere rotation, we define also
conditions of the polar vortices affecting on the stability boundaries of the
block modes.
