## Abstract

We are concerned with the system of the $N$ vortex points on a sphere with
two fixed vortex points at the both poles. This article gives a reduction method
of the system to invariant dynamical systems. It is accomplished by using the
invariance of the system with respect to the shift and the pole reversal transformations, for which the polygonal ring configuration of the $N$ vortex points
at the line of latitude, called ``$N$-ring'', remains unchanged. We prove that
there exists the $2p$-dimensional invariant dynamical system reduced by the $p$-shift transformation for arbitrary factor $p$ of $N$, and the $p$-shift invariant system is
equivalent to the $p$-vortex points system generated by the averaged Hamiltonian on
the sphere with the modified pole vortices.
It is also shown that the system can be reduced by the pole reversal transformation
when the pole vortices are identical. Since the reduced dynamical systems are defined
by the linear combination of the eigenvectors obtained in the linear stability
analysis for the $N$-ring, we obtain the inclusion structure among the invariant
reduced dynamical systems, which allows us to decompose the system of the large
vortex points into a collection of small reduced systems.