Polar Morse-Smale systems with two saddles on $n$-sphere

In the paper, one constructs the examples of polar Morse-Smale systems (diffeomorphisms and flows) with a sink fixed point, source fixed point and two saddles fixed points on $n$-dimensional sphere $\mathbb{S}^n$, $n\geq 3$. To prove this result, we construct different decompositions of the $n$-dimensional sphere $\mathbb{S}^n$. Moreover, the Morse index of a saddle fixed point can be any value between $1$ and $n-1$, and the Morse indexes of the saddles fixed points are always different. One proves that the unstable manifold of the saddle fixed point with the biggest Morse index is (transversally) intersected with the stable manifold of another saddle fixed point.