The authors in this interesting paper discuss various aspects of the phase portrait - including limit cycles, saddle points, perturbations, bifurcations, etc. of the not yet fully understood problem of cubic vector fields. They specifically consider the cubic vector field \(P(x,y)=\partial H(x,y)/\partial y\), \(Q(x,y)=-\partial H(x,y)/\partial x\), where the associated Hamiltonian \(H\) is given by \(H=[\alpha (x+h)^ 2+by^ 2-1][\alpha (x-h)^ 2+by^ 2-1]\). Earlier studies by R. Bamon and others established the result that the maximum number of limit cycles of quadratic vector fields be finite - a problem sometimes known as the Hilbert’s sixteenth problem. The authors discuss fairly extensively various crucial knowledge of the phase portrait of the above special cubic vector fields and obtain new results about the limit cycles, saddle points, perturbations of these vector fields, bifurcations of graphs, etc. Their results continue the earlier works of P. J. Holmes, D. A. Rand, Li Ji-bin; and that of N. G. Lloyd, T. R. Blows and M. C. Kalenge [Nonlinearity 1, No. 4, 653-669 (1988; Zbl 0691.34024)] on cubic vector fields and their limit cycles, etc.