On two conjectures concerning quadratic differential systems (Q1377219)

The author proved the following two conjectures of mine. Conjecture I. For the quadratic system: \[ \dot x= -y+\ell x^2+ mxy+ ny^2,\quad \dot y= x(1+ ax+ by)\tag{1} \] if \((n+b)(n+ \ell)^2- a^2(n+ b+ z\ell)= 0\), and \(m\neq a(b+ z\ell)/(n+ \ell)\), \(m<0\), then (1) has no limit cycle around \(O(0,0)\). Conjecture II. For the system: \[ \dot x= -y+\delta x+\ell x^2+ ny^2,\quad \dot y= x(1+ ax-y)\tag{2} \] if \(-\ell< na^2< (n-1)(n+ \ell)^2\), \(a^2> 4(n-1)(1- \ell)\), then (2) cannot simultaneously have limit cycles around \(O\) and another focus lying on \(1+ ax- y=0\), respectively. The case of Conjecture I when \(m>0\) is not proved.