Isn't the drawn kripke structure satisfied S1?

Question

Isn't it satisfy S1 formula and not satisfy S2? 

Answer 1

In your transition system, there is only one path that alternates infinitely often between the states labeled with {} and {p,q}. On this path, both formulas S1 and S2 are satisfied, since we have there infinitely often p&q, infinitely often p, and infinitely often q.