The distribution of limit cycles in quadratic systems with four finite singularities (Q1283966)

Consider the system \((*)\) \(dx/dt = P_2 (x,y)\), \(dy/dt = Q_2 (x,y)\) where \(P_2\) and \(Q_2\) are polynomials in \(x\) and \(y\) of degree two. It is assumed that \((*)\) has exactly four finite equilibria. Then, if the limit cycles of \((*)\) are distributed over two nests, each nest contains exactly one limit cycle. Under the additional assumption that one of these four equilibria is a weak focus (the divergence of the vector field vanishes at this equilibrium) then \((*)\) has at most one limit cycle not surrounding the weak focus.