The authors find all possible phase portraits, up to topological equivalence, of vector fields of the form \(\dot x = H_y\), \(\dot y = -H_x\), where the Hamiltonian has the form \(H = \frac12 y^2 + P(x) / Q(x)\) and \(P\) and \(Q\) are polynomials of degree at most two. The tools used are a change of coordinates to put \(H\) in a standard form, the time-rescaling by \(Q^2\) in order to obtain a polynomial system, which while no longer Hamiltonian still admits the first integral \(H\), the Poincaré compactification to obtain a vector field on the compact manifold \(S^2\), resolution of degenerate singularities by means of blowing up, and the theorem that topological equivalence is determined by the separatrix structure. They also give bifurcation diagrams that show how the different phase portraits are arranged in parameter space.