Abelian integrals of quadratic Hamiltonian vector fields with an invariant straight line (Q1915429)

The authors consider a special case of the weakened 16th Hilbert problem, that is to find the number \(N(n, m)\) of isolated zeros of the Abelian integrals \[ I(h)= \int_{\Gamma_h} Q(x, y) dx- P(x, y) dy, \] where \(\Gamma_h\) varies in the compact components of \(H^{- 1}(h)\), and \(H(x, y)\) is a real polynomial of degree \(n+ 1\), \(Q(x, y)\), \(P(x, y)\) are real polynomials of degree at most \(m\). The weakened 16th Hilbert problem is closely related to the problem of determining an upper bound of the number of limit cycles for the perturbed Hamiltonian system \[ \dot x= {\partial H\over \partial y}+ \varepsilon P(x, y),\quad \dot y= - {\partial H\over \partial x}+ \varepsilon Q(x, y). \] The authors prove that \(N(2, 2)= 1\) for the quadratic perturbation of quadratic Hamiltonian vector fields having an invariant straight line. They also give a classification of all phase portraits of such systems.